Sequence and SeriesPYQ Jan 26Question 4567 of 150
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A GP series consists of 2n terms. If the sum of the terms occupying the odd places is S1\displaystyle S_1 and that of the terms in even places is S2\displaystyle S_2, the common ratio of the progression is?

Options

An
B2S1\displaystyle 2S_1
CS2S1\displaystyle \frac{S_2}{S_1}
DS1S2\displaystyle \frac{S_1}{S_2}
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Correct Answer

✅ Option c — S2S1\displaystyle \frac{S_2}{S_1}

All Options:

  • An
  • B2S1\displaystyle 2S_1
  • CS2S1\displaystyle \frac{S_2}{S_1}
  • DS1S2\displaystyle \frac{S_1}{S_2}

Detailed Solution & Explanation

Let the Geometric Progression (GP) be: a,ar,ar2,ar3,…,ar2n−1a, ar, ar^2, ar^3, \dots, ar^{2n-1} where a\displaystyle a is the first term, r\displaystyle r is the common ratio, and the total number of terms is 2n\displaystyle 2n.
The terms occupying the odd places are: t1,t3,t5,…,t2n−1  ⟹  a,ar2,ar4,…,ar2n−2t_1, t_3, t_5, \dots, t_{2n-1} \implies a, ar^2, ar^4, \dots, ar^{2n-2} This is a GP of n\displaystyle n terms with first term a\displaystyle a and common ratio r2\displaystyle r^2. The sum of these odd-placed terms is S1\displaystyle S_1: S1=a((r2)n−1)r2−1=a(r2n−1)r2−1— (1)S_1 = \frac{a((r^2)^n - 1)}{r^2 - 1} = \frac{a(r^{2n} - 1)}{r^2 - 1} \quad \text{--- (1)}
The terms occupying the even places are: t2,t4,t6,…,t2n  ⟹  ar,ar3,ar5,…,ar2n−1t_2, t_4, t_6, \dots, t_{2n} \implies ar, ar^3, ar^5, \dots, ar^{2n-1} This is a GP of n\displaystyle n terms with first term ar\displaystyle ar and common ratio r2\displaystyle r^2. The sum of these even-placed terms is S2\displaystyle S_2: S2=ar((r2)n−1)r2−1=ar(r2n−1)r2−1— (2)S_2 = \frac{ar((r^2)^n - 1)}{r^2 - 1} = \frac{ar(r^{2n} - 1)}{r^2 - 1} \quad \text{--- (2)}
Dividing equation (2) by equation (1): S2S1=ar(r2n−1)r2−1a(r2n−1)r2−1=r\frac{S_2}{S_1} = \frac{\frac{ar(r^{2n} - 1)}{r^2 - 1}}{\frac{a(r^{2n} - 1)}{r^2 - 1}} = r Thus, the common ratio of the GP is: r=S2S1r = \frac{S_2}{S_1} Hence, **Option C** 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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