Let A_n= ( 3 4 )- ( 3 4 )^2+ ( 3 4 )^3- +(-1)^ n-1 ( 3 4 )^n and B_n=1-A_n . Then, the least odd natural number p , so that B_n>A_n , for all n p is

Step-by-step solution

A_n= ( 3 4 )- ( 3 4 )^2+ ( 3 4 )^3- +(-1)^ n-1 ( 3 4 )^n Which is a G.P. with a= 3 4 , r= -3 4 and number of terms =n aligned A_n &= 3 4 (1- ( -3 4 )^n ) 1- ( -3 4 ) = 3 4 (1- ( -3 4 )^n ) 7 4 A_n= 3 7 [1- ( -3 4 )^n ] As, B_n=1-A_n aligned For least odd natural number p , such that B_n>A_n 1-A_n>A_n 1>2 A_n A_n ( 3 4 )^n ( 1 6 )=n ( 3 4 ) 6.228 < n Hence, n should be 7 .