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) th term is $\frac{1 }{ r^n}.$

Answer 1

Let a be the first term and r be the common ratio of the G.P.
Sum of first n terms =$\frac{ a(1 - r^n) }{ (1 - r)}$
Since there are n terms from (n +1) th to (2n) th term,
Sum of terms from (n + 1)th to (2n) th term $= a_{n + 1}\frac{ (1 - r^n) }{ (1 - r)}$
a n +1 = ar n + 1-1 = arn
Thus, required ratio$=\frac{ a(1 - r^n) }{ (1 - r) }×\frac{ (1 - r) }{ ar^n (1 - r^n)} = \frac{1 }{r^n}$
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 $\frac{1 }{r^n}.$