For each positive integer n, let A_ n = ( array l n r array ) 0 r n . Then the number of elements n is 1,2,3, . .20 for which 1.9 A_ n A_ n-1 2 is :

Step-by-step solution

A_ n = ( n r ) case-I n=2 k( even ) A_ n = 2 k k n-1=2 k-1 (odd) A_ n-1 = 2 k-1 k-1 or 2 k-1 k 1.9 A _ n A _ n -1 2 1.9 ^ 2 k C _ k ^ 2 k -1 C _ k -1 2 1.9 2 2 which is true for every n= even n=2,4,6, . .20 10 numbers case-II n=(2 k-1) (odd) A_ n = 2 k-1 k or ^ 2 k-1 C_ k-1 A_ n-1 = ^ n k -2 C_ k-1 1.9 A _ n A _ n -1 21.9 ^ 2 k -1 C _ k ^ 2 k -2 C _ k -1 2.01.9 2 k-1 k 21.9 k 2 k -1 2 k 10 k n 19 aligned n &=19 only total &=11 aligned