Sequence and SeriesMCQPYQ Dec 23Question 1837 of 212
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Given an infinite geometric series with first term 'a\displaystyle a' and common ratio 'r\displaystyle r'. If its sum is 4 and the second term is 34\displaystyle \frac{3}{4}, then one of correct option is

Options

Aa=1\displaystyle a=1 and r=34\displaystyle r=\frac{3}{4}
Ba=3\displaystyle a=3 and r=14\displaystyle r=\frac{1}{4}
Ca=3\displaystyle a=3 and r=14\displaystyle r=\frac{1}{4}
Da=1\displaystyle a=1 and r=12\displaystyle r=\frac{1}{2}
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Correct Answer

✅ Option c — a=3\displaystyle a=3 and r=14\displaystyle r=\frac{1}{4}

All Options:

  • Aa=1\displaystyle a=1 and r=34\displaystyle r=\frac{3}{4}
  • Ba=3\displaystyle a=3 and r=14\displaystyle r=\frac{1}{4}
  • Ca=3\displaystyle a=3 and r=14\displaystyle r=\frac{1}{4}
  • Da=1\displaystyle a=1 and r=12\displaystyle r=\frac{1}{2}

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Detailed Solution & Explanation

The sum to infinity of a G.P. is:
S∞=a1−r=4  ⟹  a=4(1−r)S_{\infty} = \frac{a}{1-r} = 4 \implies a = 4(1-r)
The second term is:
t2=ar=34t_2 = ar = \frac{3}{4}
Substitute a=4(1−r)\displaystyle a = 4(1-r):
4(1−r)r=34  ⟹  16r(1−r)=34(1-r)r = \frac{3}{4} \implies 16r(1-r) = 3
16r2−16r+3=016r^2 - 16r + 3 = 0
Factorizing:
(4r−1)(4r−3)=0  ⟹  r=14 or r=34(4r - 1)(4r - 3) = 0 \implies r = \frac{1}{4} \text{ or } r = \frac{3}{4}

If r=14\displaystyle r = \frac{1}{4}, then a=4(1−1/4)=3\displaystyle a = 4(1 - 1/4) = 3.
If r=34\displaystyle r = \frac{3}{4}, then a=4(1−3/4)=1\displaystyle a = 4(1 - 3/4) = 1.
Option C matches a=3\displaystyle a=3 and r=14\displaystyle r=\frac{1}{4}.
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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