Question

Show that `2^(4n+4)-5n-15 ,w h e r en in N` is divisible by 225.

Answer 1

We have,
`2^(4n+4)-15n-16`
`= 2^(4(n+1))-15n-16`
`= 16^(n-1)-15n -16`
`= (1+15)^(n+1)-15n-16`
`= .^(n+1)C_(0)15^(0) + .^(n+1)C_(1)15^(1)+.^(n+1)C_(2)15^(2)+.^(n+1)C_(3)15+"…."+.^(n+1)C_(n+1)(15)^(n+1)-15n-16`
`= 1+(n+1)15+.^(n+1)C_(2)15^(2)+.^(n+1)C_(3)15^(3)+"....."+.^(n+1)C_(n+1)(15)^(n+1)-15n-16`
`= 15^(2)[.^(n+1)C_(2)+.^(n+1)C_(3)15+"....."]`
Thus, `2^(4n+4) - 15n -16` is divisible by 225.