Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from

(n+1)th to (2n)th term is $$\dfrac{1}{r^n}$$

Asked by Pragya Singh | 1 year ago |  68

##### Solution :-

Let a be the first term and r be the common ratio of the G.P.

$$s_n=\dfrac{a(1-r^n)}{1-r}$$

Since there are n terms from (n +1)th to (2n)th term,

Sum of terms from(n + 1)th to (2n)th term

$$s_n=\dfrac{a_{n+1}(1-r^n)}{1-r}$$

n +1 = ar n + 1 – 1 = arn

Thus, required ratio =

$$\dfrac{a(1-r^n)}{1-r}\times \dfrac{1-r}{ar^n(1-r^n)}$$

$$\dfrac{1}{r^n}$$

Thus, the ratio of the sum of first n terms of a G.P. to the sum of terms from

(n + 1)th to (2n)th term is $$\dfrac{1}{r^n}$$

Answered by Abhisek | 1 year ago

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