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NCERT Class XI Mathematics - Sequences and Series - Solutions

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Question : 58 of 106
Marks: +1, -0
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 1rn\frac{1}{r^n}.
Solution:  
Let the G.P. be a, ar, ar2ar^2, .......
Sum of first n terms = a + ar +...... + arn−1ar^{n-1} ⇒ SnS_n = a(1−rn)1−r\frac{a(1-r^n)}{1-r}
Let the sum of terms from (n + 1)th term to (2n)th term denoted by
Sn′S_n' = arn+arn+1ar^n+ar^{n+1} + ... + ar2n−1ar^{2n-1}
∴ Sn′S_n' = arn(1−rn)1−r\frac{ar^n(1-r^n)}{1-r}. Now , SnSn′\frac{S_n}{S_n'} = a(1−rn)1−rarn(1−rn)1−r\frac{\frac{a(1-r^n)}{1-r}}{\frac{ar^n(1-r^n)}{1-r}} = 1rn\frac{1}{r^n}
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