`1+5+9+...+(4n-3)=n(2n-1)` for all natural numbers `n`

prove that 1+5+9+ . . .+(4n-3)=n(2n-1), for all natural number n.

prove that 1+5+9+ . . .+(4n-3)=n(2n-1), for all natural number n.

Prove that : (2n) ! = 2^n (n!)[1.3.5.... (2n-1)] for all natural numbers n.

Prove that 1+2+2^(2)+ . . .+2^(n)=2^(n+1)-1 , for all natural number n.

Prove that 1+2+2^(2)+ . . .+2^(n)=2^(n+1)-1 , for all natural number n.

If A = [(1,1),(1,1)] , prove by induction that A^n = [(2^(n-1), 2^(n-1)), (2^(n-1), 2^(n-1))] for all natural numbers n.

Prove that , (1)/(n+1) + (1)/(n+2) + ………….+ (1)/(2n) gt 13/24 for all natural numbers n > 1

Show that ^nP_n= "^nP_(n-1) for all natural numbers n .

Let x_(n)=(2^(n)+3^(n))^(1//2n) for all natural number n. Then

Prove by the principle of mathematical induction, that 1.1!+2.2!+3.3!+.....+(n.n!)=(n+1)!-1"for all natural number "n (n!=1xx2xx3....n)