Question

Show that the ratio of the sum of first "n" terms of a G.P. to the sum of terms from
${(n + 1)}^{th} to \: (2n) \: term \: is \: \frac{1}{ {r}^{n} }$
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Answer 1

Step-by-step explanation:

We have,

Sum of first n terms where, first term = 'a' and common ratio = 'r'

$s_{n} = \frac{a( {r}^{n} - 1)}{(r - 1)}$

Now, we can write the sum of (n + 1)th term to (2n)th term as follows

$s_{(2n)} - s_{(n)} = \frac{a( {r}^{2n} - 1)}{(r - 1)} - \frac{a( {r}^{n} - 1) }{(r - 1)}$

$= \frac{ar^{2n} - ar^{n} }{(r - 1)}$

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

Hence, the required ratio

$\frac{ \frac{a( {r}^{n} - 1) }{(r - 1)} }{ \frac{ a {r}^{n}( {r}^{n} - 1)}{(r - 1)}}$

$= \frac{1}{ {r}^{n} }$