Question

Find the sum to infinity of the given arithmetico-geometric sequences.
$1,\frac{4}{3}, \frac{7}{9}, \frac{10}{27}, \frac{13}{81}, \frac{16}{243},...$

Answer 1

Solution :

Given arithmetico-geometric sequences.
$1,\frac{4}{3}, \frac{7}{9}, \frac{10}{27}, \frac{13}{81}, \frac{16}{243},...$

********************************†****

We know that ,

Sum of infinite terms of an AGP

= a/( 1 - r ) + ( dr )/( 1 - r² )

Where | r | < 1
****************************************

Here ,

Sum of Infinite AGP :

1 + 4/3 + 7/9 + 10/27 + 13/81 +....

a = 1 , d = 3 , r = 1/3

Sum of the infinite series

= 1/( 1 - 1/3 )[ ( 3 × 1/3 )/[ 1 - 1/3² ]

= 1/(2/3) { 1/( 8/9 ) ]

= ( 3/2 )( 9/8 )

= 27/16

••••