The third term of a GP is 3. What is the product of the first five terms?

This question was previously asked in
NDA (Held On: 22 April 2018) Maths Previous Year paper
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  1. 216
  2. 226
  3. 243
  4. Cannot be determined due to insufficient data

Answer (Detailed Solution Below)

Option 3 : 243
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Detailed Solution

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Concepts:

Let us consider sequence a1, a2, a3 …. an is an G.P.

  • Common ratio = r = \(\frac{{{a_2}}}{{{a_1}}} = \frac{{{a_3}}}{{{a_2}}} = \ldots = \frac{{{a_n}}}{{{a_{n - 1}}}}\)
  • nth term of the G.P. is an = arn−1
  • Sum of n terms = s = \(\frac{{a\;\left( {{r^n} - 1} \right)}}{{r - 1}}\); where r >1
  • Sum of n terms = s = \(\frac{{a\;\left( {1 - {r^n}} \right)}}{{1 - r}}\); where r <1
  • Sum of infinite GP = \({{\rm{s}}_\infty } = {\rm{}}\frac{{\rm{a}}}{{1{\rm{}} - {\rm{r}}}}{\rm{}}\); |r| < 1

Where a is 1st term and r is common ratio.

Calculation:

Given: The third term of a GP is 3

Let 'a' be the first term and 'r' be the common ratio.

∴ T3 = ar2 = 3

We know that Tn = a rn-1

So, T1 = a, T2 = ar, T3 = ar2, T4 = ar3, T5 = ar4

Now, Product of the first five terms = a × ar × ar2 × ar3 × ar4 = a5r10 = (ar2)5 = 35 = 243
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