Prove that following identities:
P(n,n) = P(n,n-1)

Prove the following identity : 2u_6-3u_4+1=0 , where u_n =cos^n theta+ sin^n theta .

Prove the following by using induction for all n in N . 1+2+3+.....+n=(n(n+1))/(2)

By the Principle of Mathematical Induction, prove the following for all n in N : 4^(n)-3n-1 is divisible by 9.

By the Principle of Mathematical Induction, prove the following for all n in N : 4^(n)+ 15n-1 is divisible by 9.

If x^p occurs in the expansion of (x^2+1/x)^(2n) , prove that its coefficient is : ((2n)!)/([1/3(4n-p)!][1/3(2n+p)!]) .

Prove that the Principle of Mathematical Induction does not apply to the following : P(n) : " n^3+n is divisible by 3" .

Prove that the Principle of Mathematical Induction does not apply to the following : P(n) : " n^3> 100 ".

Prove the following by using the principle of mathematical induction for all n in N :- n(n +1) (n + 5) is a multiple of 3.

Prove the following by using the principle of mathematical induction for all n in N :- 1.2.3 + 2.3.4 +...+n(n+1)(n+2)=(n(n+1)(n+2)(n+3))/4 .

Give an example of the following statement : P(n) such that it is true for all n in N.