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Show that 2^(4n+4)-15n-16, where n in...

Show that `2^(4n+4)-15n-16, where ` n ` in N` is divisible by 225.

Text Solution

Verified by Experts

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.
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