Sequence and SeriesMCQPYQ Jun 22, MTP Dec 22 - Series IQuestion 1855 of 212
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The sum of first eight terms of geometric progression is five times the sum of the first four terms. The common ratio is

Options

A3\displaystyle \sqrt{3}
B2\displaystyle \sqrt{2}
C4\displaystyle 4
D2\displaystyle 2
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Correct Answer

✅ Option b — 2\displaystyle \sqrt{2}

All Options:

  • A3\displaystyle \sqrt{3}
  • B2\displaystyle \sqrt{2}
  • C4\displaystyle 4
  • D2\displaystyle 2

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Detailed Solution & Explanation

We are given that S8=5⋅S4\displaystyle S_8 = 5 \cdot S_4.
Using the GP sum formula:
a(r8−1)r−1=5⋅a(r4−1)r−1\frac{a(r^8 - 1)}{r-1} = 5 \cdot \frac{a(r^4 - 1)}{r-1}
r8−1=5(r4−1)(assuming r≠1)r^8 - 1 = 5(r^4 - 1) \quad (\text{assuming } r \neq 1)
(r4−1)(r4+1)=5(r4−1)(r^4 - 1)(r^4 + 1) = 5(r^4 - 1)
r4+1=5  ⟹  r4=4  ⟹  r2=2  ⟹  r=±2r^4 + 1 = 5 \implies r^4 = 4 \implies r^2 = 2 \implies r = \pm \sqrt{2}
Hence, **Option B** is the correct answer.

About This Chapter: Sequence and Series

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Arithmetic & Geometric Progressions

This chapter covers Arithmetic Progressions (AP) and Geometric Progressions (GP). Students learn how to find the nth term, sum of n terms, arithmetic/geometric means, and sum to infinity of a GP.

View Official ICAI Syllabus

Exam Strategy Tip

For complex 'sum of series' questions, a great hack is to substitute n = 1 and n = 2 into the question and the options to see which one matches, completely bypassing the formula.

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