Solution 1: To prove for an integer.

Solution 1: To prove for an integer ,
Consider left hand side
Simplify further we get,
Hence, . This completes the proof.
Now, let’s simplify .
We know that,
Differentiating with respect to x we get,
Substitute we get,
Solution a: Suppose that is an integer such that . This implies
that
And
For some integers
Subtract equation (1) from equation (2) we get,
Suppose we get,
Since, and are the two consecutive integers and we know that product of two
consecutive integer is divisible by 2. That is must be multiple of 2, and hence
.
Solution b: To prove that for an even integer n.