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Sequences and Series question

2019 · 9 Jan · Shift 2 · Q24
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Sequences and Series question

2019 · 9 Jan · Shift 2 · Q24

JEE MainMathematicsSequences and SeriesMCQ+4 / −1
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 ac{a \over c}ca​ equal to :
  1. A
    2
  2. B
    12{1 \over 2}21​
  3. C
    713{7 \over 13}137​
  4. D
    4
View written solutionFree

Correct answer: D

  1. Write the A.P. terms in terms of first term and common difference

Let the first term of the A.P. be AAA and common difference be ddd.

Then the nnnth term is Tn=A+(n−1)d.T_n = A + (n-1)d.Tn​=A+(n−1)d.

So, a=T7=A+6d,a = T_7 = A+6d,a=T7​=A+6d, b=T11=A+10d,b = T_{11} = A+10d,b=T11​=A+10d, c=T13=A+12d.c = T_{13} = A+12d.c=T13​=A+12d.

Since the A.P. is non-constant, we have d≠0.d \ne 0.d=0.

  1. Use the condition that a,b,ca,b,ca,b,c are consecutive terms of a G.P.

For three consecutive terms of a G.P., the middle term squared equals the product of the other two: b2=ac.b^2 = ac.b2=ac.

Substitute the values: (A+10d)2=(A+6d)(A+12d).(A+10d)^2 = (A+6d)(A+12d).(A+10d)2=(A+6d)(A+12d).

  1. Expand both sides

Left side: (A+10d)2=A2+20Ad+100d2.(A+10d)^2 = A^2 + 20Ad + 100d^2.(A+10d)2=A2+20Ad+100d2.

Right side: (A+6d)(A+12d)=A2+18Ad+72d2.(A+6d)(A+12d) = A^2 + 18Ad + 72d^2.(A+6d)(A+12d)=A2+18Ad+72d2.

Equating, A2+20Ad+100d2=A2+18Ad+72d2.A^2 + 20Ad + 100d^2 = A^2 + 18Ad + 72d^2.A2+20Ad+100d2=A2+18Ad+72d2.

Cancel A2A^2A2: 20Ad+100d2=18Ad+72d2.20Ad + 100d^2 = 18Ad + 72d^2.20Ad+100d2=18Ad+72d2.

So, 2Ad+28d2=0.2Ad + 28d^2 = 0.2Ad+28d2=0.

Factor: 2d(A+14d)=0.2d(A+14d)=0.2d(A+14d)=0.

Since d≠0d \ne 0d=0, we get A+14d=0⇒A=−14d.A+14d=0 \quad \Rightarrow \quad A=-14d.A+14d=0⇒A=−14d.

  1. Find aaa and ccc

Now, a=A+6d=−14d+6d=−8d,a = A+6d = -14d+6d = -8d,a=A+6d=−14d+6d=−8d, c=A+12d=−14d+12d=−2d.c = A+12d = -14d+12d = -2d.c=A+12d=−14d+12d=−2d.

Therefore, ac=−8d−2d=4.\frac{a}{c} = \frac{-8d}{-2d} = 4.ca​=−2d−8d​=4.

  1. Check options

Thus, ac=4.\frac{a}{c}=4.ca​=4.

So the correct option is D.

  1. Comparison with stored answer

Stored correct answer: D

Derived answer: D

They agree.

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