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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 1 by r raised to n

24.  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 \frac{1}{r^n}

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Let first term =a  and common ratio = r.

sum \, \, of\, \, n\, \, terms=\frac{a(1-r^n)}{1-r}

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

Sum of terms from (n+1) to 2n.

S_n=\frac{a_(_n+_1_)(1-r^n)}{1-r}

a_(_n+_1)=a.r^{n+1-1}=ar^n

Thus, the required ratio  = \frac{a(1-r^n)}{1-r}\times \frac{1-r}{ar^n(1-r^n)}

                                =\frac{1}{r^n}

Thus,  the common 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  \frac{1}{r^n}.
 

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