Sequence and SeriesPYQ Jan 26Question 4511 of 150
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The third term of a geometric progression is 5. Then the product of first five terms is

Options

A55\displaystyle 5^5
B56\displaystyle 5^6
C57\displaystyle 5^7
D59\displaystyle 5^9
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Correct Answer

✅ Option a — 55\displaystyle 5^5

All Options:

  • A55\displaystyle 5^5
  • B56\displaystyle 5^6
  • C57\displaystyle 5^7
  • D59\displaystyle 5^9

Detailed Solution & Explanation

Let the first term of the Geometric Progression (GP) be a\displaystyle a and the common ratio be r\displaystyle r.
The general term of a GP is given by:
tn=arn−1t_n = a r^{n-1}

Given that the third term (t3\displaystyle t_3) is 5\displaystyle 5:
t3=ar3−1=ar2=5t_3 = a r^{3-1} = a r^2 = 5 ---(Equation 1)

We need to find the product of the first five terms of this GP:
Product=t1×t2×t3×t4×t5\text{Product} = t_1 \times t_2 \times t_3 \times t_4 \times t_5
Product=a×(ar)×(ar2)×(ar3)×(ar4)\text{Product} = a \times (ar) \times (ar^2) \times (ar^3) \times (ar^4)Product=a5r1+2+3+4\text{Product} = a^5 r^{1+2+3+4}
Product=a5r10\text{Product} = a^5 r^{10}
Product=(ar2)5\text{Product} = (a r^2)^5

Substitute the value of ar2=5\displaystyle a r^2 = 5 from Equation 1:
Product=(5)5=55\text{Product} = (5)^5 = 5^5

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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