Prove that following identities:
`P(n,n) = lfloorrP(n,n-r)`

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)

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

Let R be a relation from N to N defined by R = {(a,b): a , b in N and a =b^(3)} . Are the following true ? (a, a) in R, " for all " a in N

By the Principle of Mathematical Induction, prove the following for all n in N : 1+4+7+.....+ (3n-2)= (n(3n-1))/2 .

Prove the following by using the principle of mathematical induction for all n in N :- a + ar + ar^2+...+ ar^(n-1)=(a(r^n-1))/(r-1) .

For the proposition P(n), given by , 1+3+5+.........+(2n-1) = n^2 +2 , prove that P(k) is true implies P(k + 1) is true. But, P(n) is not true for all n in N.

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 .

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