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Fifth term of a GP is 2, then the product of its first 9 terms is
1. 256
2. 512
3. 1024
4. None of these
5. 64

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Correct Answer - Option 2 : 512

Concepts:

Let us consider the sequence a1, a2, a3 …. an is a 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:

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

We know that Tn = a rn-1

Given: ar4 = 2

Now, Product of 9 terms = a × ar × ar2 × ar3 × ar4 × ar5 × ar6 × ar7 × ar8

= a9 r36 = (ar4)9 = 29 = 512

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