Logo
Jeeify
MCQMediumJEE 2024Geometric Progression (GP)

JEE Mathematics 2024 Question with Solution

If each term of a geometric progression a1,a2,a3,…a_1, a_2, a_3, \ldots with a1=18a_1 = \frac{1}{8} and a2≠a1a_2 \ne a_1, is the arithmetic mean of the next two terms and Sn=a1+a2+…+anS_n = a_1 + a_2 + \ldots + a_n, then S20−S18S_{20} - S_{18} is equal to:

  • A

    2152^{15}

  • B

    −218-2^{18}

  • C

    2182^{18}

  • D

    −215-2^{15}

Answer

Correct answer:D

Step-by-step solution

Standard Method

Given: The sequence is a geometric progression with a1=18a_1 = \frac{1}{8} and each term is the arithmetic mean of the next two terms.

Find: S20−S18S_{20} - S_{18}.

Let the common ratio be rr. Then the general term is

an=a1rn−1a_n = a_1 r^{n-1}

and the condition

an=an+1+an+22a_n = \frac{a_{n+1} + a_{n+2}}{2}

gives

2an=an+1+an+22a_n = a_{n+1} + a_{n+2}

Detailed Algebra

Substitute the GP terms:

2arn−1=arn+arn+12ar^{n-1} = ar^n + ar^{n+1}

Dividing by arn−1ar^{n-1}, we get

2=r+r22 = r + r^2

so

r2+r−2=0r^2 + r - 2 = 0

which factorises as

(r−1)(r+2)=0(r-1)(r+2)=0

Hence

r=1 or r=−2r=1 \text{ or } r=-2

Since a2≠a1a_2 \ne a_1, we must have r≠1r \ne 1. Therefore,

r=−2r=-2

Now,

S20−S18=a19+a20S_{20} - S_{18} = a_{19} + a_{20}

Using an=18(−2)n−1a_n = \frac{1}{8}(-2)^{n-1},

a19=18(−2)18,a20=18(−2)19a_{19} = \frac{1}{8}(-2)^{18}, \qquad a_{20} = \frac{1}{8}(-2)^{19}

Therefore,

S20−S18=18[(−2)18+(−2)19]S_{20} - S_{18} = \frac{1}{8}\left[(-2)^{18} + (-2)^{19}\right] =18(−2)18(1−2)= \frac{1}{8}(-2)^{18}(1-2) =−18⋅218= -\frac{1}{8} \cdot 2^{18} =−215= -2^{15}

Therefore, the correct option is D.

The first approach in the source solution leads to a wrong quadratic and inconsistent simplification. The second approach correctly derives r=−2r=-2, which matches the marked correct option.

Common mistakes

  • Using the condition incorrectly as r=r2+r3r = r^2 + r^3. The relation must come from 2an=an+1+an+22a_n = a_{n+1}+a_{n+2}, which simplifies to 2=r+r22 = r + r^2 after dividing by the common GP factor.

  • Forgetting the condition a2≠a1a_2 \ne a_1. This excludes r=1r=1, so after solving the quadratic you must choose r=−2r=-2.

  • Applying the sum formula unnecessarily and making algebra errors. Since S20−S18=a19+a20S_{20} - S_{18} = a_{19}+a_{20}, it is faster and safer to use the last two terms directly.

Practice more Geometric Progression (GP) questions

Get unlimited AI-adaptive practice, mastery tracking, and an AI tutor that explains every step — free to start.

Start free trialMore on Geometric Progression (GP)

Related questions