Question

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

Answer 1

Let us consider the first term of G.P is a and the common ratio is r.

Now, the sum of the G.P when, r<1 is,

Sum of first n terms= a( 1− r n ) ( 1−r )

Now, there are n terms from ( n+1 ) th to ( 2n ) th term, then, the sum of series from ( n+1 ) th to ( 2n ) th when, r<1 is,

Sum of terms from ( n+1 ) th to ( 2n ) th term= a n+1 ( 1− r n ) 1−r

Now, the ratio of sum of the n term of G.P and sum of n term form ( n+1 ) th to ( 2n ) th

Required ratio = a( 1− r n ) ( 1−r ) a r n ( 1− r n ) ( 1−r ) = a( 1− r n ) ( 1−r ) × ( 1−r ) a r n ( 1− r n ) = a( 1− r n ) a r n ( 1− r n ) = 1 r n

Hence, it is proved that the ratio of the sum of the n term of G.P and sum of n term the form ( n+1 ) th to ( 2n ) th is 1 r n .