Sequence and SeriesMCQMTP Nov 20Question 1877 of 212
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The sum of the first two terms of an infinite geometric series is 15\displaystyle 15 and each term is equal to the sum of all the terms following it, then the sum of the series is

Options

A20\displaystyle 20
B15\displaystyle 15
C25\displaystyle 25
DNone
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Correct Answer

✅ Option a — 20\displaystyle 20

All Options:

  • A20\displaystyle 20
  • B15\displaystyle 15
  • C25\displaystyle 25
  • DNone

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Detailed Solution & Explanation

Let the infinite geometric progression be a,ar,ar2,…\displaystyle a, ar, ar^2, \dots
We are given:
1) Sum of first two terms is 15\displaystyle 15:
a+ar=15  ⟹  a(1+r)=15— (1)a + ar = 15 \implies a(1+r) = 15 \quad \text{--- (1)}
2) Each term equals the sum of all following terms:
tn=∑k=n+1∞tkt_n = \sum_{k=n+1}^{\infty} t_k
arn−1=arn1−rar^{n-1} = \frac{ar^n}{1-r}
1=r1−r  ⟹  1−r=r  ⟹  2r=1  ⟹  r=121 = \frac{r}{1-r} \implies 1-r = r \implies 2r = 1 \implies r = \frac{1}{2}

Substitute r=12\displaystyle r = \frac{1}{2} into equation (1):
a(1+12)=15  ⟹  a(32)=15  ⟹  a=10a\left(1 + \frac{1}{2}\right) = 15 \implies a\left(\frac{3}{2}\right) = 15 \implies a = 10

The sum of the series is:
S∞=a1−r=101−1/2=20S_{\infty} = \frac{a}{1-r} = \frac{10}{1 - 1/2} = 20
Hence, **Option A** 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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