New chat PYQ Library Doubts Rank report

Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →

Sequences and Series question

2015 · Shift 2 · Q28

Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.

PYQ Library
JEE Advanced
Mathematics
Sequences and Series
2015 · Shift 2 · Q28

Sequences and Series question

2015 · Shift 2 · Q28

JEE AdvancedMathematicsSequences and SeriesNumerical+4 / −1

The coefficient of

{x^9}

in the expansion of (1 + x) (1 +

{x^2)}

(1 +

{x^3}

) ....

(1 + {x^{100}})

is

Numerical answer

View written solutionFree

Correct answer: 8

We need the coefficient of $x^9$ in

(1+x)(1+x^2)(1+x^3) \cdots (1+x^{100}).

In the product $\prod_{k=1}^{100}(1+x^k)$ , choosing either $1$ or $x^k$ from each factor means:

every term corresponds to selecting some distinct numbers from $1,2,3,\dots,100$ ,
the exponent of $x$ is the sum of the selected numbers.

So, the coefficient of $x^9$ equals the number of ways to write $9$ as a sum of distinct positive integers.

Since any part bigger than $9$ cannot be used, we only need partitions of $9$ into distinct parts.

List all partitions of $9$ into distinct positive integers:

9

8+1

7+2

6+3

6+2+1

5+4

5+3+1

4+3+2

These are all possible distinct-part partitions of $9$ .

Therefore, the coefficient of $x^9$ is

8.

Comparing with the stored correct answer: stored answer = $8$ , which matches our result.

More from Sequences and Series

Let a, b, c be positive integers such that $\frac{b}{a}$ is an integer. If a, b, c are in geometric progression and the arithmetic mean of a, b, c is b + 2, then the value of $\frac{a ^{2} + a - 14}{a + 1}$ is2014 · Numerical
A pack contains $n$ cards numbered from $1$ to $n .$ Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is $1224.$ If the smaller of the numbers on the removed cards is $k,$ then $k - 20 =$ …2013 · Numerical
Let $S_{n} = k = 1 \sum 4 n (- 1)^{\frac{k ( k + 1 )}{2}} k^{2} .$ Then $S_{n}$ can take value(s)2013 · Multiple correct
Let $a_{1}, a_{2}, a_{3}, .....$ be in harmonic progression with $a_{1} = 5$ and $a_{20} = 25.$ The least positive integer $n$ for which $a_{n} < 0$ is2012 · MCQ
Let $a_{1}$ , $a_{2}$ , $a_{3}$ ........ $a_{100}$ be an arithmetic progression with $a_{1}$ = 3 and $S_{p} = i = 1 \sum p a_{i}, 1 \leq p \leq 100$ . For any integer n with $1 \leq n \leq 20$ , let m = 5n. If $\frac{S _{m}}{S _{n}}$ …2011 · Numerical
Let $S_{k}$ = 1, 2,....., 100, denote the sum of the infinite geometric series whose first term is $\frac{k - 1}{k !}$ and the common ratio is $\frac{1}{k}$ . Then the value of $\frac{100 ^{2}}{100 !} + k = 1 \sum 100 (k^{2} - 3 k + 1) S_{k}$ …2010 · Numerical
Let $a_{1}, a_{2}, a_{3}$ ......, $a_{11}$ be real numbers satisfying $a_{1} = 15, 27 - 2 a_{2} > 0 an d a_{k} = 2 a_{k - 1} - a_{k - 2} f or k = 3, 4, ........11$ . if $\frac{a _{1}^{2} + a _{2}^{2} + .... + a _{11}^{2}}{11} = 90$ …2010 · Numerical
If the sum of first $n$ terms of an A.P. is $c n^{2}$ , then the sum of squares of these $n$ terms is2009 · MCQ