Sequences and Series

181 questions · Mathematics · JEE Main

Sequences and Series

181 questions · Mathematics · JEE Main

Let $a_{1}, a_{2}, a_{3}, \dots$ be in an A.P. such that $\sum_{k = 1}^{12} a_{2 k - 1} = - \frac{72}{5} a_{1}, a_{1} e q 0$ . If $\sum_{k = 1}^{n} a_{k} = 0$ , then $n$ is :2025 · 2 Apr · Shift 1 · Q45 · MCQ
The number of terms of an A.P. is even; the sum of all the odd terms is 24 , the sum of all the even terms is 30 and the last term exceeds the first by $\frac{21}{2}$ . Then the number of terms which are integers in the A.P. is :2025 · 2 Apr · Shift 2 · Q39 · MCQ
If the sum of the first 10 terms of the series $\frac{4 \cdot 1}{1 + 4 \cdot 1 ^{4}} + \frac{4 \cdot 2}{1 + 4 \cdot 2 ^{4}} + \frac{4 \cdot 3}{1 + 4 \cdot 3 ^{4}} + \dots .$ . is $\frac{m}{n}$ , where $gcd (m, n) = 1$ …2025 · 2 Apr · Shift 2 · Q48 · Numerical
Let $a_{1}, a_{2}, a_{3}, \dots$ . be a G.P. of increasing positive numbers. If $a_{3} a_{5} = 729$ and $a_{2} + a_{4} = \frac{111}{4}$ , then $24 (a_{1} + a_{2} + a_{3})$ is equal to2025 · 3 Apr · Shift 1 · Q27 · MCQ
The sum $1 + 3 + 11 + 25 + 45 + 71 + \dots$ upto 20 terms, is equal to2025 · 3 Apr · Shift 1 · Q28 · MCQ
The sum $1 + \frac{1 + 3}{2 !} + \frac{1 + 3 + 5}{3 !} + \frac{1 + 3 + 5 + 7}{4 !} + \dots$ upto $\infty$ terms, is equal to2025 · 3 Apr · Shift 2 · Q44 · MCQ
Let $A = {1, 6, 11, 16, \dots}$ and $B = {9, 16, 23, 30, \dots}$ be the sets consisting of the first 2025 terms of two arithmetic progressions. Then $n (A \cup B)$ is2025 · 4 Apr · Shift 1 · Q28 · MCQ
$1 + 3 + 5^{2} + 7 + 9^{2} + \dots$ upto 40 terms is equal to2025 · 4 Apr · Shift 1 · Q32 · MCQ
If the sum of the first 20 terms of the series $\frac{4 \cdot 1}{4 + 3 \cdot 1 ^{2} + 1 ^{4}} + \frac{4 \cdot 2}{4 + 3 \cdot 2 ^{2} + 2 ^{4}} + \frac{4 \cdot 3}{4 + 3 \cdot 3 ^{2} + 3 ^{4}} + \frac{4 \cdot 4}{4 + 3 \cdot 4 ^{2} + 4 ^{4}} + \dots \cdot$ is…2025 · 4 Apr · Shift 2 · Q33 · MCQ
Consider two sets A and B, each containing three numbers in A.P. Let the sum and the product of the elements of A be 36 and p respectively and the sum and the product of the elements of B be 36 and $q$ respectively. Let d and D be the…2025 · 4 Apr · Shift 2 · Q43 · MCQ
Let $x_{1}, x_{2}, x_{3}, x_{4}$ be in a geometric progression. If $2, 7, 9, 5$ are subtracted respectively from $x_{1}, x_{2}, x_{3}, x_{4}$ , then the resulting numbers are in an arithmetic progression. Then the value of $\frac{1}{24} (x_{1} x_{2} x_{3} x_{4})$ …2025 · 7 Apr · Shift 1 · Q45 · MCQ
Let $a_{n}$ be the $n^{t h}$ term of an A.P. If $S_{n} = a_{1} + a_{2} + a_{3} + \dots + a_{n} = 700$ , $a_{6} = 7$ and $S_{7} = 7$ , then $a_{n}$ is equal to :2025 · 7 Apr · Shift 2 · Q29 · MCQ
If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum of its first nine terms is :2025 · 7 Apr · Shift 2 · Q36 · MCQ
If $\frac{1}{1 ^{4}} + \frac{1}{2 ^{4}} + \frac{1}{3 ^{4}} + \dots \infty = \frac{π ^{4}}{90}$ , $\frac{1}{1 ^{4}} + \frac{1}{3 ^{4}} + \frac{1}{5 ^{4}} + \dots \infty = α$ , $\frac{1}{2 ^{4}} + \frac{1}{4 ^{4}} + \frac{1}{6 ^{4}} + \dots \infty = β$ , then $\frac{α}{β}$ …2025 · 8 Apr · Shift 2 · Q28 · MCQ
Let $a_{1}, a_{2}, a_{3}, \dots$ be a G.P. of increasing positive terms. If $a_{1} a_{5} = 28$ and $a_{2} + a_{4} = 29$ , then $a_{6}$ is equal to:2025 · 22 Jan · Shift 1 · Q38 · MCQ
Suppose that the number of terms in an A.P. is $2 k, k \in N$ . If the sum of all odd terms of the A.P. is 40 , the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27 , then k is equal to:2025 · 22 Jan · Shift 2 · Q43 · MCQ
If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to2025 · 23 Jan · Shift 1 · Q41 · MCQ
The roots of the quadratic equation $3 x^{2} - p x + q = 0$ are $1 0^{th}$ and $1 1^{th}$ terms of an arithmetic progression with common difference $\frac{3}{2}$ . If the sum of the first 11 terms of this arithmetic progression is…2025 · 23 Jan · Shift 2 · Q49 · Numerical
Let $S_{n} = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \dots$ upto $n$ terms. If the sum of the first six terms of an A.P. with first term -p and common difference p is $2026 S_{2025}$ , then the absolute difference…2025 · 24 Jan · Shift 1 · Q41 · MCQ
In an arithmetic progression, if $S_{40} = 1030$ and $S_{12} = 57$ , then $S_{30} - S_{10}$ is equal to :2025 · 24 Jan · Shift 2 · Q26 · MCQ
If $7 = 5 + \frac{1}{7} (5 + α) + \frac{1}{7 ^{2}} (5 + 2 α) + \frac{1}{7 ^{3}} (5 + 3 α) + \dots\dots\dots\dots \infty$ , then the value of $α$ is :2025 · 24 Jan · Shift 2 · Q36 · MCQ
Let $⟨ a_{n} ⟩$ be a sequence such that $a_{0} = 0, a_{1} = \frac{1}{2}$ and $2 a_{n + 2} = 5 a_{n + 1} - 3 a_{n}, n = 0, 1, 2, 3, \dots$ . Then $k = 1 \sum 100 a_{k}$ is equal to2025 · 28 Jan · Shift 1 · Q30 · MCQ
Let $T_{r}$ be the $r^{th}$ term of an A.P. If for some $m, T_{m} = \frac{1}{25}, T_{25} = \frac{1}{20}$ , and $20 r = 1 \sum 25 T_{r} = 13$ …2025 · 28 Jan · Shift 1 · Q33 · MCQ
For positive integers $n$ , if $4 a_{n} = (n^{2} + 5 n + 6)$ and $S_{n} = k = 1 \sum n (\frac{1}{a _{k}})$ , then the value of $507 S_{2025}$ is :2025 · 28 Jan · Shift 2 · Q45 · MCQ
The interior angles of a polygon with n sides, are in an A.P. with common difference 6°. If the largest interior angle of the polygon is 219°, then n is equal to .2025 · 28 Jan · Shift 2 · Q49 · Numerical
Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11th term is :2025 · 29 Jan · Shift 1 · Q29 · MCQ
Let $a_{1}, a_{2}, \dots, a_{2024}$ be an Arithmetic Progression such that $a_{1} + (a_{5} + a_{10} + a_{15} + \dots + a_{2020}) + a_{2024} = 2233$ . Then $a_{1} + a_{2} + a_{3} + \dots + a_{2024}$ is equal to .2025 · 29 Jan · Shift 2 · Q50 · Numerical
Let $3, a, b, c$ be in A.P. and $3, a - 1, b + 1, c + 9$ be in G.P. Then, the arithmetic mean of $a, b$ and $c$ is :2024 · 1 Feb · Shift 1 · Q47 · MCQ
Let $3, 7, 11, 15, \dots, 403$ and $2, 5, 8, 11, \dots, 404$ be two arithmetic progressions. Then the sum, of the common terms in them, is equal to .2024 · 1 Feb · Shift 1 · Q54 · Numerical
Let $S_{n}$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{10} = 390$ and the ratio of the tenth and the fifth terms is $15 : 7$ , then $S_{15} - S_{5}$ is equal to :2024 · 1 Feb · Shift 2 · Q48 · MCQ
If three successive terms of a G.P. with common ratio $r (r > 1)$ are the lengths of the sides of a triangle and $[r]$ denotes the greatest integer less than or equal to $r$ , then $3 [r] + [- r]$ is equal to …2024 · 1 Feb · Shift 2 · Q59 · Numerical
Let the first three terms 2, p and q, with $q e q 2$ , of a G.P. be respectively the $7^{th}, 8^{th}$ and $1 3^{th}$ terms of an A.P. If the $5^{th}$ term of the G.P. is the $n^{th}$ term of the…2024 · 4 Apr · Shift 1 · Q44 · MCQ
The value of $\frac{1 \times 2 ^{2} + 2 \times 3 ^{2} + \dots + 100 \times ( 101 ) ^{2}}{1 ^{2} \times 2 + 2 ^{2} \times 3 + \dots . + 10 0 ^{2} \times 101}$ is2024 · 4 Apr · Shift 2 · Q33 · MCQ
Let three real numbers $a, b, c$ be in arithmetic progression and $a + 1, b, c + 3$ be in geometric progression. If $a > 10$ and the arithmetic mean of $a, b$ and $c$ is 8, then the cube of the geometric mean of $a, b$ and $c$ is2024 · 4 Apr · Shift 2 · Q47 · MCQ
If $\frac{1}{1 + 2} + \frac{1}{2 + 3} + \dots + \frac{1}{99 + 100} = m$ and $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \dots + \frac{1}{99 \cdot 100} = n$ , then the point $(m, n)$ lies…2024 · 5 Apr · Shift 1 · Q42 · MCQ
Let $a_{1}, a_{2}, a_{3}, \dots$ be in an arithmetic progression of positive terms. Let $A_{k} = a_{1}^{2} - a_{2}^{2} + a_{3}^{2} - a_{4}^{2} + \dots + a_{2 k - 1}^{2} - a_{2 k}^{2}$ . If $A_{3} = - 153, A_{5} = - 435$ and $a_{1}^{2} + a_{2}^{2} + a_{3}^{2} = 66$ …2024 · 5 Apr · Shift 1 · Q57 · Numerical
For $x ⩾ 0$ , the least value of $K$ , for which $4^{1 + x} + 4^{1 - x}, \frac{K}{2}, 1 6^{x} + 1 6^{- x}$ are three consecutive terms of an A.P., is equal to :2024 · 5 Apr · Shift 2 · Q46 · MCQ
If $1 + \frac{3 - 2}{2 3} + \frac{5 - 2 6}{18} + \frac{9 3 - 11 2}{36 3} + \frac{49 - 20 6}{180} + \dots$ upto $\infty = 2 + (\frac{b}{a} + 1) lo g_{e} (\frac{a}{b})$ , where a…2024 · 5 Apr · Shift 2 · Q55 · Numerical
Let the first term of a series be $T_{1} = 6$ and its $r^{th}$ term $T_{r} = 3 T_{r - 1} + 6^{r}, r = 2, 3$ , ............ $n$ . If the sum of the first $n$ terms of this series is $\frac{1}{5} (n^{2} - 12 n + 39) (4 \cdot 6^{n} - 5 \cdot 3^{n} + 1)$ …2024 · 6 Apr · Shift 1 · Q56 · Numerical
Let $A B C$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $A B C$ and the same process is repeated infinitely many times. If $P$ is the sum of perimeters and $Q$ is…2024 · 6 Apr · Shift 2 · Q41 · MCQ
A software company sets up m number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took…2024 · 6 Apr · Shift 2 · Q43 · MCQ
If $S (x) = (1 + x) + 2 (1 + x)^{2} + 3 (1 + x)^{3} + \dots + 60 (1 + x)^{60}, x e q 0$ , and $(60)^{2} S (60) = a (b)^{b} + b$ , where $a, b \in N$ , then $(a + b)$ equal to .2024 · 6 Apr · Shift 2 · Q58 · Numerical
Let the positive integers be written in the form : If the $k^{th}$ row contains exactly $k$ numbers for every natural number $k$ , then the row in which the number 5310 will be, is . Includes diagram2024 · 8 Apr · Shift 1 · Q54 · Numerical
Let $α = \sum_{r = 0}^{n} (4 r^{2} + 2 r + 1)^{n} C_{r}$ and $β = (\sum_{r = 0}^{n} \frac{^{n} C _{r}}{r + 1}) + \frac{1}{n + 1}$ . If $140 < \frac{2 α}{β} < 281$ , then the value of $n$ is .2024 · 8 Apr · Shift 1 · Q57 · Numerical
In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $\frac{70}{3}$ and the product of the third and fifth terms is 49. Then the sum of the $4^{th}, 6^{th}$ and $8^{th}$ …2024 · 8 Apr · Shift 2 · Q41 · MCQ
An arithmetic progression is written in the following way The sum of all the terms of the 10th row is . Includes diagram2024 · 8 Apr · Shift 2 · Q59 · Numerical
If the sum of the series $\frac{1}{1 \cdot ( 1 + d )} + \frac{1}{( 1 + d ) ( 1 + 2 d )} + \dots + \frac{1}{( 1 + 9 d ) ( 1 + 10 d )}$ is equal to 5, then $50 d$ is equal to :2024 · 9 Apr · Shift 1 · Q37 · MCQ
Let $a, a r, a r^{2}$ , ............ be an infinite G.P. If $\sum_{n = 0}^{\infty} a r^{n} = 57$ and $\sum_{n = 0}^{\infty} a^{3} r^{3 n} = 9747$ , then $a + 18 r$ is equal to2024 · 9 Apr · Shift 2 · Q48 · MCQ
If $(\frac{1}{α + 1} + \frac{1}{α + 2} + \dots .. + \frac{1}{α + 1012}) - (\frac{1}{2 \cdot 1} + \frac{1}{4 \cdot 3} + \frac{1}{6 \cdot 5} + \dots\dots + \frac{1}{2024 \cdot 2023}) = \frac{1}{2024}$ , then $α$ is…2024 · 9 Apr · Shift 2 · Q60 · Numerical
The number of common terms in the progressions $4, 9, 14, 19, \dots\dots$ , up to $2 5^{th}$ term and $3, 6, 9, 12, \dots\dots$ , up to $3 7^{th}$ term is :2024 · 27 Jan · Shift 1 · Q46 · MCQ
If $8 = 3 + \frac{1}{4} (3 + p) + \frac{1}{4 ^{2}} (3 + 2 p) + \frac{1}{4 ^{3}} (3 + 3 p) + \dots\dots \infty$ , then the value of $p$ is .2024 · 27 Jan · Shift 1 · Q51 · Numerical
$The 2 0^{th} term from the end of the progression 20, 19 \frac{1}{4}, 18 \frac{1}{2}, 17 \frac{3}{4}, \dots, - 129 \frac{1}{4} is :$ 2024 · 27 Jan · Shift 2 · Q39 · MCQ
If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to2024 · 29 Jan · Shift 1 · Q35 · MCQ
In an A.P., the sixth term $a_{6} = 2$ . If the product $a_{1} a_{4} a_{5}$ is the greatest, then the common difference of the A.P. is equal to2024 · 29 Jan · Shift 1 · Q37 · MCQ
If $lo g_{e} a, lo g_{e} b, lo g_{e} c$ are in an A.P. and $lo g_{e} a - lo g_{e} 2 b, lo g_{e} 2 b - lo g_{e} 3 c, lo g_{e} 3 c - lo g_{e}$ a are also in an A.P, then $a : b : c$ …2024 · 29 Jan · Shift 2 · Q41 · MCQ
If each term of a geometric progression $a_{1}, a_{2}, a_{3}, \dots$ with $a_{1} = \frac{1}{8}$ and $a_{2} e q a_{1}$ , is the arithmetic mean of the next two terms and $S_{n} = a_{1} + a_{2} + \dots .. + a_{n}$ , then $S_{20} - S_{18}$ is equal to2024 · 29 Jan · Shift 2 · Q46 · MCQ
Let $S_{n}$ denote the sum of first $n$ terms of an arithmetic progression. If $S_{20} = 790$ and $S_{10} = 145$ , then $S_{15} - S_{5}$ is :2024 · 30 Jan · Shift 1 · Q49 · MCQ
Let $α = 1^{2} + 4^{2} + 8^{2} + 1 3^{2} + 1 9^{2} + 2 6^{2} + \dots$ upto 10 terms and $β = \sum_{n = 1}^{10} n^{4}$ . If $4 α - β = 55 k + 40$ , then $k$ is equal to .2024 · 30 Jan · Shift 1 · Q58 · Numerical
Let $a$ and $b$ be be two distinct positive real numbers. Let $1 1^{th}$ term of a GP, whose first term is $a$ and third term is $b$ , is equal to $p^{th}$ term of another GP, whose first term is $a$ and fifth term is $b$ .…2024 · 30 Jan · Shift 2 · Q32 · MCQ
Let $S_{n}$ be the sum to $n$ -terms of an arithmetic progression $3, 7, 11$ , If $40 < (\frac{6}{n ( n + 1 )} \sum_{k = 1}^{n} S_{k}) < 42$ , then $n$ equals .2024 · 30 Jan · Shift 2 · Q60 · Numerical
For $0 < c < b < a$ , let $(a + b - 2 c) x^{2} + (b + c - 2 a) x + (c + a - 2 b) = 0$ and $α \neq = 1$ be one of its root. Then, among the two statements (I) If $α \in (- 1, 0)$ , then $b$ cannot be the geometric mean of $a$ and $c$ (II) If $α \in (0, 1)$ …2024 · 31 Jan · Shift 1 · Q35 · MCQ
The sum of the series $\frac{1}{1 - 3 \cdot 1 ^{2} + 1 ^{4}} + \frac{2}{1 - 3 \cdot 2 ^{2} + 2 ^{4}} + \frac{3}{1 - 3 \cdot 3 ^{2} + 3 ^{4}} + \dots$ up to 10 -terms is2024 · 31 Jan · Shift 1 · Q40 · MCQ
Let $2^{nd}, 8^{th}$ and $4 4^{th}$ terms of a non-constant A. P. be respectively the $1^{st}, 2^{nd}$ and $3^{rd}$ terms of a G. P. If the first term of the A. P. is 1, then the sum of…2024 · 31 Jan · Shift 2 · Q44 · MCQ
Let $a_{1} = 8, a_{2}, a_{3}, \dots, a_{n}$ be an A.P. If the sum of its first four terms is 50 and the sum of its last four terms is 170 , then the product of its middle two terms is .2023 · 1 Feb · Shift 1 · Q36 · Numerical
The sum of the common terms of the following three arithmetic progressions. $3, 7, 11, 15, \dots ., 399$ , $2, 5, 8, 11, \dots ., 359$ and $2, 7, 12, 17, \dots ., 197$ , is equal to .2023 · 1 Feb · Shift 2 · Q43 · Numerical
If $(20)^{19} + 2 (21) (20)^{18} + 3 (21)^{2} (20)^{17} + \dots + 20 (21)^{19} = k (20)^{19}$ , then $k$ is equal to .2023 · 6 Apr · Shift 2 · Q43 · Numerical
Let $S_{K} = \frac{1 + 2 + \dots + K}{K}$ and $\sum_{j = 1}^{n} S_{j}^{2} = \frac{n}{A} (B n^{2} + C n + D)$ , where $A, B, C, D \in N$ and $A$ has least value. Then2023 · 8 Apr · Shift 1 · Q35 · MCQ
Let the first term $α$ and the common ratio r of a geometric progression be positive integers. If the sum of squares of its first three terms is 33033, then the sum of these three terms is equal to2023 · 10 Apr · Shift 1 · Q37 · MCQ
Let $x_{1}, x_{2}, \dots, x_{100}$ be in an arithmetic progression, with $x_{1} = 2$ and their mean equal to 200 . If $y_{i} = i (x_{i} - i), 1 \leq i \leq 100$ , then the mean of $y_{1}, y_{2}, \dots, y_{100}$ is :2023 · 11 Apr · Shift 1 · Q27 · MCQ
Let $a, b, c$ and $d$ be positive real numbers such that $a + b + c + d = 11$ . If the maximum value of $a^{5} b^{3} c^{2} d$ is $3750 β$ , then the value of $β$ is2023 · 11 Apr · Shift 2 · Q23 · MCQ
Let $s_{1}, s_{2}, s_{3}, \dots, s_{10}$ respectively be the sum to 12 terms of 10 A.P. s whose first terms are $1, 2, 3, \dots .10$ and the common differences are $1, 3, 5, \dots\dots, 19$ respectively. Then $\sum_{i = 1}^{10} s_{i}$ is…2023 · 13 Apr · Shift 1 · Q31 · MCQ
Let a $_{1}$ , a $_{2}$ , a $_{3}$ , .... be a G.P. of increasing positive numbers. Let the sum of its 6th and 8th terms be 2 and the product of its 3rd and 5th terms be $\frac{1}{9}$ . Then $6 (a_{2} + a_{4}) (a_{4} + a_{6})$ is equal to2023 · 13 Apr · Shift 2 · Q21 · MCQ
Let $A_{1}$ and $A_{2}$ be two arithmetic means and $G_{1}, G_{2}, G_{3}$ be three geometric means of two distinct positive numbers. Then $G_{1}^{4} + G_{2}^{4} + G_{3}^{4} + G_{1}^{2} G_{3}^{2}$ is equal to :2023 · 15 Apr · Shift 1 · Q35 · MCQ
For three positive integers p, q, r, $x^{p q^{2}} = y^{q r} = z^{p^{2} r}$ and r = pq + 1 such that 3, 3 log $_{y} x$ , 3 log $_{z} y$ , 7 log $_{x} z$ are in A.P. with common difference $\frac{1}{2}$ . Then r-p-q is equal to2023 · 24 Jan · Shift 1 · Q34 · MCQ
The 4 $^{th}$ term of GP is 500 and its common ratio is $\frac{1}{m}, m \in N$ . Let $S_{n}$ denote the sum of the first n terms of this GP. If $S_{6} > S_{5} + 1$ and $S_{7} < S_{6} + \frac{1}{2}$ , then…2023 · 24 Jan · Shift 1 · Q39 · Numerical
For the two positive numbers $a, b,$ if $a, b$ and $\frac{1}{18}$ are in a geometric progression, while $\frac{1}{a}, 10$ and $\frac{1}{b}$ are in an arithmetic progression, then $16 a + 12 b$ is equal to .2023 · 25 Jan · Shift 2 · Q44 · Numerical
Let $a_{1}, a_{2}, a_{3}, ...$ be a $GP$ of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24, then $a_{1} a_{9} + a_{2} a_{4} a_{9} + a_{5} + a_{7}$ is equal to .2023 · 29 Jan · Shift 1 · Q41 · Numerical
Let ${a_{k}}$ and ${b_{k}}, k \in N$ , be two G.P.s with common ratios $r_{1}$ and $r_{2}$ respectively such that $a_{1} = b_{1} = 4$ and $r_{1} < r_{2}$ . Let $c_{k} = a_{k} + b_{k}, k \in N$ . If $c_{2} = 5$ and $c_{3} = \frac{13}{4}$ …2023 · 29 Jan · Shift 2 · Q43 · Numerical
Let $a, b, c > 1, a^{3}, b^{3}$ and $c^{3}$ be in A.P., and $lo g_{a} b, lo g_{c} a$ and $lo g_{b} c$ be in G.P. If the sum of first 20 terms of an A.P., whose first term is $\frac{a + 4 b + c}{3}$ and the common difference is $\frac{a - 8 b + c}{10}$ is $- 444$ …2023 · 30 Jan · Shift 2 · Q27 · MCQ
If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296 , respectively, then the sum of common ratios of all such GPs is2023 · 31 Jan · Shift 1 · Q28 · MCQ
Let $a_{1}, a_{2}, \dots, a_{n}$ be in A.P. If $a_{5} = 2 a_{7}$ and $a_{11} = 18$ , then $12 (\frac{1}{a _{10} + a _{11}} + \frac{1}{a _{11} + a _{12}} + \dots + \frac{1}{a _{17} + a _{18}})$ is equal to …2023 · 31 Jan · Shift 1 · Q44 · Numerical
Let $a_{1}, a_{2}, a_{3}, \dots$ be an A.P. If $a_{7} = 3$ , the product $a_{1} a_{4}$ is minimum and the sum of its first $n$ terms is zero, then $n! - 4 a_{n (n + 2)}$ is equal to :2023 · 31 Jan · Shift 2 · Q29 · MCQ
If ${a_{i}}_{i = 1}^{n}$ , where n is an even integer, is an arithmetic progression with common difference 1, and $i = 1 \sum n a_{i} = 192, i = 1 \sum n /2 a_{2 i} = 120$ , then n is equal to :2022 · 24 Jun · Shift 1 · Q31 · MCQ
Let x, y > 0. If x3y2 = 215, then the least value of 3x + 2y is2022 · 24 Jun · Shift 2 · Q25 · MCQ
Let $a, b$ be two non-zero real numbers. If $p$ and $r$ are the roots of the equation $x^{2} - 8 a x + 2 a = 0$ and $q$ and s are the roots of the equation $x^{2} + 12 b x + 6 b = 0$ , such that $\frac{1}{p}, \frac{1}{q}, \frac{1}{r}, \frac{1}{s}$ …2022 · 25 Jul · Shift 1 · Q39 · Numerical
Consider two G.Ps. 2, 22, 23, ..... and 4, 42, 43, .... of 60 and n terms respectively. If the geometric mean of all the 60 + n terms is $(2)^{\frac{225}{8}}$ , then $k = 1 \sum n k (n - k)$ is equal to :2022 · 26 Jul · Shift 1 · Q29 · MCQ
Different A.P.'s are constructed with the first term 100, the last term 199, and integral common differences. The sum of the common differences of all such A.P.'s having at least 3 terms and at most 33 terms is .2022 · 26 Jul · Shift 2 · Q38 · Numerical
If a1 (> 0), a2, a3, a4, a5 are in a G.P., a2 + a4 = 2a3 + 1 and 3a2 + a3 = 2a4, then a2 + a4 + 2a5 is equal to .2022 · 26 Jun · Shift 2 · Q44 · Numerical
Suppose $a_{1}, a_{2}, \dots, a_{n}$ , .. be an arithmetic progression of natural numbers. If the ratio of the sum of first five terms to the sum of first nine terms of the progression is $5 : 17$ and , $110 < a_{15} < 120$ , then the…2022 · 27 Jul · Shift 1 · Q29 · MCQ
Let the sum of an infinite G.P., whose first term is a and the common ratio is r, be 5 . Let the sum of its first five terms be $\frac{98}{25}$ . Then the sum of the first 21 terms of an AP, whose first term is $10 ar, n^{th}$ …2022 · 27 Jul · Shift 2 · Q27 · MCQ
$x = n = 0 \sum \infty a^{n}, y = n = 0 \sum \infty b^{n}, z = n = 0 \sum \infty c^{n}$ , where a, b, c are in A.P. and |a| < 1, |b| < 1, |c| < 1, abc $e$ 0, then :2022 · 27 Jun · Shift 1 · Q22 · MCQ
If a1, a2, a3 ...... and b1, b2, b3 ....... are A.P., and a1 = 2, a10 = 3, a1b1 = 1 = a10b10, then a4 b4 is equal to -2022 · 27 Jun · Shift 2 · Q24 · MCQ
Let A1, A2, A3, ....... be an increasing geometric progression of positive real numbers. If A1A3A5A7 = $\frac{1}{1296}$ and A2 + A4 = $\frac{7}{36}$ , then the value of A6 + A8 + A10 is equal to2022 · 28 Jun · Shift 1 · Q27 · MCQ
Let A = {1, a1, a2 ....... a18, 77} be a set of integers with 1 1 2 18 1 + a2 + ...... + a18 is equal to .2022 · 28 Jun · Shift 1 · Q40 · Numerical
If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is 1 : 7 and a + n = 33, then the value of n is :2022 · 28 Jun · Shift 2 · Q26 · MCQ
Let for n = 1, 2, ......, 50, Sn be the sum of the infinite geometric progression whose first term is n2 and whose common ratio is $\frac{1}{( n + 1 ) ^{2}}$ . Then the value of $\frac{1}{26} + n = 1 \sum 50 (S_{n} + \frac{2}{n + 1} - n - 1)$ …2022 · 28 Jun · Shift 2 · Q45 · Numerical
Let $a_{1}, a_{2}, a_{3}, \dots$ be an A.P. If $r = 1 \sum \infty \frac{a _{r}}{2 ^{r}} = 4$ , then $4 a_{2}$ is equal to .2022 · 29 Jul · Shift 1 · Q42 · Numerical
Let 3, 6, 9, 12, ....... upto 78 terms and 5, 9, 13, 17, ...... upto 59 terms be two series. Then, the sum of the terms common to both the series is equal to .2022 · 29 Jun · Shift 2 · Q37 · Numerical
Let a1, a2, ..........., a21 be an AP such that $n = 1 \sum 20 \frac{1}{a _{n} a _{n + 1}} = \frac{4}{9}$ . If the sum of this AP is 189, then a6a16 is equal to :2021 · 1 Sep · Shift 2 · Q36 · MCQ
Consider an arithmetic series and a geometric series having four initial terms from the set {11, 8, 21, 16, 26, 32, 4}. If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these…2021 · 16 Mar · Shift 1 · Q40 · Numerical
Let $\frac{1}{16}$ , a and b be in G.P. and $\frac{1}{a}$ , $\frac{1}{b}$ , 6 be in A.P., where a, b > 0. Then 72(a + b) is equal to .2021 · 16 Mar · Shift 2 · Q41 · Numerical
Let S1 be the sum of first 2n terms of an arithmetic progression. Let S2 be the sum of first 4n terms of the same arithmetic progression. If (S2 $-$ S1) is 1000, then the sum of the first 6n terms of the arithmetic progression is equal to :2021 · 18 Mar · Shift 2 · Q36 · MCQ
Let Sn denote the sum of first n-terms of an arithmetic progression. If S10 = 530, S5 = 140, then S20 $-$ S6 is equal to:2021 · 22 Jul · Shift 2 · Q24 · MCQ
The sum of all the elements in the set {n $\in$ {1, 2, ....., 100} | H.C.F. of n and 2040 is 1} is equal to .2021 · 22 Jul · Shift 2 · Q42 · Numerical
The sum of first four terms of a geometric progression (G. P.) is $\frac{65}{12}$ and the sum of their respective reciprocals is $\frac{65}{18}$ . If the product of first three terms of the G.P. is 1, and the third term is $α$ ,…2021 · 24 Feb · Shift 2 · Q41 · Numerical
Let A1, A2, A3, ....... be squares such that for each n $\geq$ 1, the length of the side of An equals the length of diagonal of An+1. If the length of A1 is 12 cm, then the smallest value of n for which area of An is less than one, is …2021 · 25 Feb · Shift 1 · Q41 · Numerical
The minimum value of $f (x) = a^{a^{x}} + a^{1 - a^{x}}$ , where a, $x \in R$ and a > 0, is equal to :2021 · 25 Feb · Shift 2 · Q34 · MCQ
Let Sn be the sum of the first n terms of an arithmetic progression. If S3n = 3S2n, then the value of $\frac{S _{4 n}}{S _{2 n}}$ is :2021 · 25 Jul · Shift 1 · Q25 · MCQ
If the sum of an infinite GP a, ar, ar2, ar3, ....... is 15 and the sum of the squares of its each term is 150, then the sum of ar2, ar4, ar6, ....... is :2021 · 26 Aug · Shift 1 · Q33 · MCQ
The sum of all 3-digit numbers less than or equal to 500, that are formed without using the digit "1" and they all are multiple of 11, is .2021 · 26 Aug · Shift 2 · Q39 · Numerical
Let a1, a2, ......., a10 be an AP with common difference $-$ 3 and b1, b2, ........., b10 be a GP with common ratio 2. Let ck = ak + bk, k = 1, 2, ......, 10. If c2 = 12 and c3 = 13, then $k = 1 \sum 10 c_{k}$ is equal to …2021 · 26 Aug · Shift 2 · Q42 · Numerical
In an increasing geometric series, the sum of the second and the sixth term is $\frac{25}{2}$ and the product of the third and fifth term is 25. Then, the sum of 4th, 6th and 8th terms is equal to :2021 · 26 Feb · Shift 1 · Q36 · MCQ
The total number of 4-digit numbers whose greatest common divisor with 18 is 3, is .2021 · 26 Feb · Shift 2 · Q40 · Numerical
If the arithmetic mean and geometric mean of the pth and qth terms of the sequence $-$ 16, 8, $-$ 4, 2, ...... satisfy the equation 4x2 $-$ 9x + 5 = 0, then p + q is equal to .2021 · 26 Feb · Shift 2 · Q44 · Numerical
If $lo g_{3} 2, lo g_{3} (2^{x} - 5), lo g_{3} (2^{x} - \frac{7}{2})$ are in an arithmetic progression, then the value of x is equal to .2021 · 27 Jul · Shift 1 · Q41 · Numerical
Three numbers are in an increasing geometric progression with common ratio r. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference d. If the fourth term of GP is 3 r2, then r2 $-$ d…2021 · 31 Aug · Shift 1 · Q26 · MCQ
Let a1, a2, a3, ..... be an A.P. If $\frac{a _{1} + a _{2} + .... + a _{10}}{a _{1} + a _{2} + .... + a _{p}} = \frac{100}{p ^{2}}$ , p $e$ 10, then $\frac{a _{11}}{a _{10}}$ is equal to :2021 · 31 Aug · Shift 2 · Q32 · MCQ
The number of 4-digit numbers which are neither multiple of 7 nor multiple of 3 is .2021 · 31 Aug · Shift 2 · Q38 · Numerical
The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in :2020 · 2 Sep · Shift 1 · Q26 · MCQ
If the sum of first 11 terms of an A.P., a1, a2, a3, .... is 0 (a $e$ 0), then the sum of the A.P., a1 , a3 , a5 ,....., a23 is ka1 , where k is equal to :2020 · 2 Sep · Shift 2 · Q29 · MCQ
If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. is :2020 · 3 Sep · Shift 1 · Q34 · MCQ
The value of $(0.16)^{l o g_{2.5} (\frac{1}{3} + \frac{1}{3 ^{2}} + .... t o \infty)}$ is equal to .2020 · 3 Sep · Shift 1 · Q40 · Numerical
If m arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that 4th A.M. is equal to 2nd G.M., then m is equal to .2020 · 3 Sep · Shift 2 · Q29 · Numerical
The minimum value of 2sinx + 2cosx is :2020 · 4 Sep · Shift 2 · Q28 · MCQ
Let a1, a2, ..., an be a given A.P. whose common difference is an integer and Sn = a1 + a2 + .... + an. If a1 = 1, an = 300 and 15 $\leq$ n $\leq$ 50, then the ordered pair (Sn-4, an–4) is equal to:2020 · 4 Sep · Shift 2 · Q36 · MCQ
If $3^{2 s i n 2 α - 1}$ , 14 and $3^{4 - 2 s i n 2 α}$ are the first three terms of an A.P. for some $α$ , then the sixth terms of this A.P. is:2020 · 5 Sep · Shift 1 · Q29 · MCQ
If the sum of the second, third and fourth terms of a positive term G.P. is 3 and the sum of its sixth, seventh and eighth terms is 243, then the sum of the first 50 terms of this G.P. is :2020 · 5 Sep · Shift 2 · Q21 · MCQ
Let a , b, c , d and p be any non zero distinct real numbers such that (a2 + b2 + c2)p2 – 2(ab + bc + cd)p + (b2 + c2 + d2) = 0. Then :2020 · 6 Sep · Shift 1 · Q23 · MCQ
The common difference of the A.P. b1, b2, … , bm is 2 more than the common difference of A.P. a1, a2, …, an. If a40 = –159, a100 = –399 and b100 = a70, then b1 is equal to :2020 · 6 Sep · Shift 2 · Q31 · MCQ
Five numbers are in A.P. whose sum is 25 and product is 2520. If one of these five numbers is - $\frac{1}{2}$ , then the greatest number amongst them is:2020 · 7 Jan · Shift 1 · Q34 · MCQ
Let $a_{1}$ , $a_{2}$ , $a_{3}$ ,....... be a G.P. such that $a_{1}$ < 0, $a_{1}$ + $a_{2}$ = 4 and $a_{3}$ + $a_{4}$ = 16. If $i = 1 \sum 9 a_{i} = 4 λ$ , then $λ$ is equal to:2020 · 7 Jan · Shift 2 · Q21 · MCQ
Let ƒ : R $\to$ R be such that for all x $\in$ R (21+x + 21–x), ƒ(x) and (3x + 3–x) are in A.P., then the minimum value of ƒ(x) is2020 · 8 Jan · Shift 1 · Q38 · MCQ
If the 10th term of an A.P. is $\frac{1}{20}$ and its 20th term is $\frac{1}{10}$ , then the sum of its first 200 terms is2020 · 8 Jan · Shift 2 · Q33 · MCQ
The number of terms common to the two A.P.'s 3, 7, 11, ....., 407 and 2, 9, 16, ....., 709 is .2020 · 9 Jan · Shift 2 · Q22 · Numerical
Let an be the nth term of a G.P. of positive terms. $n = 1 \sum 100 a_{2 n + 1} = 200$ and $n = 1 \sum 100 a_{2 n} = 100$ , then $n = 1 \sum 200 a_{n}$ is equal to :2020 · 9 Jan · Shift 2 · Q40 · MCQ
The sum of all natural numbers 'n' such that 100 < n < 200 and H.C.F. (91, n) > 1 is :2019 · 8 Apr · Shift 1 · Q27 · MCQ
If three distinct numbers a, b, c are in G.P. and the equations ax2 + 2bx + c = 0 and dx2 + 2ex + ƒ = 0 have a common root, then which one of the following statements is correct?2019 · 8 Apr · Shift 2 · Q29 · MCQ
Let the sum of the first n terms of a non-constant A.P., a1, a2, a3, ..... be $50 n + \frac{n ( n - 7 )}{2} A$ , where A is a constant. If d is the common difference of this A.P., then the ordered pair (d, a50) is equal to2019 · 9 Apr · Shift 1 · Q39 · MCQ
If the sum and product of the first three term in an A.P. are 33 and 1155, respectively, then a value of its 11th term is :-2019 · 9 Apr · Shift 2 · Q26 · MCQ
Let $a_{1}, a_{2}, ......., a_{30}$ be an A.P., $S = i = 1 \sum 30 a_{i}$ and $T = i = 1 \sum 15 a_{(2 i - 1)}$ . If $a_{5}$ = 27 and S - 2T = 75, then $a_{10}$ is equal to :2019 · 9 Jan · Shift 1 · Q23 · MCQ
If a, b, c be three distinct real numbers in G.P. and a + b + c = xb , then x cannot be2019 · 9 Jan · Shift 1 · Q35 · MCQ
Let a, b and c be the 7th, 11th and 13th terms respectively of a non-constant A.P. If these are also three consecutive terms of a G.P., then $\frac{a}{c}$ equal to :2019 · 9 Jan · Shift 2 · Q24 · MCQ
If a1, a2, a3, ............... an are in A.P. and a1 + a4 + a7 + ........... + a16 = 114, then a1 + a6 + a11 + a16 is equal to :2019 · 10 Apr · Shift 1 · Q33 · MCQ
Let a1, a2, a3,......be an A.P. with a6 = 2. Then the common difference of this A.P., which maximises the product a1a4a5, is :2019 · 10 Apr · Shift 2 · Q29 · MCQ
Let $a$ , b and c be in G.P. with common ratio r, where $a e$ 0 and 0 < r $\leq \frac{1}{2}$ . If 3 $a$ , 7b and 15c are the first three terms of an A.P., then the 4th term of this A.P. is :2019 · 10 Apr · Shift 2 · Q35 · MCQ
The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is -2019 · 10 Jan · Shift 1 · Q38 · MCQ
Let a1, a2, a3, ..... a10 be in G.P. with ai > 0 for i = 1, 2, ….., 10 and S be the set of pairs (r, k), r, k $\in$ N (the set of natural numbers) for which $lo g_{e} a_{1}^{r} a_{2}^{k} lo g_{e} a_{4}^{r} a_{5}^{k} lo g_{e} a_{7}^{r} a_{8}^{k} lo g_{e} a_{2}^{r} a_{3}^{k} lo g_{e} a_{5}^{r} a_{6}^{k} lo g_{e} a_{8}^{r} a_{9}^{k} lo g_{e} a_{3}^{r} a_{4}^{k} lo g_{e} a_{6}^{r} a_{7}^{k} lo g_{e} a_{9}^{r} a_{10}^{k}$ …2019 · 10 Jan · Shift 2 · Q36 · MCQ
The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is $\frac{27}{19}$ .Then the common ratio of this series is :2019 · 11 Jan · Shift 1 · Q35 · MCQ
Let a1, a2, . . . . . ., a10 be a G.P. If $\frac{a _{3}}{a _{1}} = 25,$ then $\frac{a _{9}}{a _{5}}$ equals2019 · 11 Jan · Shift 1 · Q40 · MCQ
Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression $\frac{x ^{m} y ^{n}}{( 1 + x ^{2 m} ) ( 1 + y ^{2 n} )}$ is :2019 · 11 Jan · Shift 2 · Q21 · MCQ
If 19th term of a non-zero A.P. is zero, then its (49th term) : (29th term) is :2019 · 11 Jan · Shift 2 · Q33 · MCQ
Let Sn denote the sum of the first n terms of an A.P. If S4 = 16 and S6= – 48, then S10 is equal to :2019 · 12 Apr · Shift 1 · Q34 · MCQ
If a1, a2, a3, ..... are in A.P. such that a1 + a7 + a16 = 40, then the sum of the first 15 terms of this A.P. is :2019 · 12 Apr · Shift 2 · Q30 · MCQ
The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an A.P. Then the sum of the original three terms of the given G.P. is :2019 · 12 Jan · Shift 1 · Q44 · MCQ
If nC4, nC5 and nC6 are in A.P., then n can be :2019 · 12 Jan · Shift 2 · Q30 · MCQ
If sin4 $α$ + 4 cos4 $β$ + 2 = 4 $2$ sin $α$ cos $β$ ; $α$ , $β \in$ [0, $π$ ], then cos( $α$ + $β$ ) $-$ cos( $α - β$ ) is equal to :2019 · 12 Jan · Shift 2 · Q44 · MCQ
If x1, x2, . . ., xn and $\frac{1}{h _{1}}$ , $\frac{1}{h _{2}}$ , . . . , $\frac{1}{h _{n}}$ are two A.P..s such that x3 = h2 = 8 and x8 = h7 = 20, then x5.h10 equals :2018 · 15 Apr · Shift 1 · Q27 · MCQ
If b is the first term of an infinite G.P. whose sum is five, then b lies in the interval :2018 · 15 Apr · Shift 1 · Q33 · MCQ
If a, b, c are in A.P. and a2, b2, c2 are in G.P. such that a < b < c and a + b + c = $\frac{3}{4},$ then the value of a is :2018 · 15 Apr · Shift 2 · Q23 · MCQ
Let $\frac{1}{x _{1}}, \frac{1}{x _{2}}, ..., \frac{1}{x _{n}}$ (xi $e$ 0 for i = 1, 2, ..., n) be in A.P. such that x1=4 and x21 = 20. If n is the least positive integer for which $x_{n} > 50,$ then $i = 1 \sum n (\frac{1}{x _{i}})$ …2018 · 16 Apr · Shift 1 · Q31 · MCQ
Let $a_{1}$ , $a_{2}$ , $a_{3}$ , ......... , $a_{49}$ be in A.P. such that $k = 0 \sum 12 a_{4 k + 1} = 416$ and $a_{9} + a_{43} = 66$ . $a_{1}^{2} + a_{2}^{2} + ....... + a_{17}^{2} = 140 m$ , then m is equal to2018 · Shift 0 · Q33 · MCQ
If the arithmetic mean of two numbers a and b, a > b > 0, is five times their geometric mean, then $\frac{a + b}{a - b}$ is equal to :2017 · 8 Apr · Shift 1 · Q27 · MCQ
If three positive numbers a, b and c are in A.P. such that abc = 8, then the minimum possible value of b is :2017 · 9 Apr · Shift 1 · Q24 · MCQ
Let x, y, z be positive real numbers such that x + y + z = 12 and x3y4z5 = (0.1) (600)3. Then x3 + y3 + z3is equal to :2016 · 9 Apr · Shift 1 · Q22 · MCQ
Let a1, a2, a3, . . . . . . . , an, . . . . . be in A.P. If a3 + a7 + a11 + a15 = 72, then the sum of its first 17 terms is equal to :2016 · 10 Apr · Shift 1 · Q21 · MCQ
If A > 0, B > 0 and A + B = $\frac{π}{6}$ , then the minimum value of tanA + tanB is :2016 · 10 Apr · Shift 1 · Q30 · MCQ
If the $2^{n d}, 5^{t h} an d 9^{t h}$ terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is :2016 · Shift 0 · Q34 · MCQ
If m is the A.M. of two distinct real numbers l and n $(l, n > 1)$ and $G_{1}, G_{2}$ and $G_{3}$ are three geometric means between $l$ and n, then $G_{1}^{4} + 2 G_{2}^{4} + G_{3}^{4}$ equals:2015 · Shift 0 · Q39 · MCQ
Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. then the common ratio of the G.P. is :2014 · Shift 0 · Q39 · MCQ
A man saves ₹ 200 in each of the first three months of his service. In each of the subsequent months his saving increases by ₹ 40 more than the saving of immediately previous month. His total saving from the start of service will be ₹…2011 · Shift 0 · Q45 · MCQ
A person is to count 4500 currency notes. Let $a_{n}$ denote the number of notes he counts in the $n^{t h}$ minute. If $a_{1}$ = $a_{2}$ = ....= $a_{10}$ = 150 and $a_{10}$ , $a_{11}$ ,.... are in an AP with common difference - 2, then the…2010 · Shift 0 · Q44 · MCQ
The first two terms of a geometric progression add up to 12. the sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term is2008 · Shift 0 · Q48 · MCQ
In a geometric progression consisting of positive terms, each term equals the sum of the next two terns. Then the common ratio of its progression is equals2007 · Shift 0 · Q57 · MCQ
If $a_{1}, a_{2}, .... a_{n}$ are in H.P., then the expression $a_{1} a_{2} + a_{2} a_{3} + .... + a_{n - 1} a_{n}$ is equal to2006 · Shift 0 · Q74 · MCQ
Let $a_{1}$ , $a_{2}$ , $a_{3}$ .....be terms on A.P. If $\frac{a _{1} + a _{2} + ..... a _{p}}{a _{1} + a _{2} + ..... a _{q}} = \frac{p ^{2}}{q ^{2}}, p e q, t h e n \frac{a _{6}}{a _{21}}$ equals2006 · Shift 0 · Q75 · MCQ
If $x = n = 0 \sum \infty a^{n}, y = n = 0 \sum \infty b^{n}, z = n = 0 \sum \infty c^{n},$ where a, b, c are in A.P and $∣ a ∣ < 1, ∣ b ∣ < 1, ∣ c ∣ < 1$ …2005 · Shift 0 · Q102 · MCQ
Let $T_{r}$ be the rth term of an A.P. whose first term is a and common difference is d. If for some positive integers m, n, $m e n, T_{m} = \frac{1}{n} an d T_{n} = \frac{1}{m},$ then a - d equals2004 · Shift 0 · Q94 · MCQ
If 1, $lo g_{9} (3^{1 - x} + 2), lo g_{3} (4. 3^{x} - 1)$ are in A.P. then x equals2002 · Shift 0 · Q110 · MCQ
l, m, n are the $p^{t h}$ , $q^{t h}$ and $r^{t h}$ term of a G.P all positive, $t h e n lo g l lo g m lo g n p q r 111 e q u a l s$ 2002 · Shift 0 · Q111 · MCQ
Fifth term of a GP is 2, then the product of its 9 terms is2002 · Shift 0 · Q112 · MCQ
Sum of infinite number of terms of GP is 20 and sum of their square is 100. The common ratio of GP is2002 · Shift 0 · Q113 · MCQ