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

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 n^(2)lt2^(n) , for all natural number n≥5 .

prove that 2nlt(n+2)! for all natural numbers n.

Prove that 2n +1 lt 2^(n) for all natural numbers n ge3

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)

prove that 2+4+6+…2n=n^(2)+n , for all natural numbers n.

Prove that 7 is a factor of 2^(3n)-1 for all natural numbers n.

Show that 10^(2n-1)+1 is divisible by 11 for all natural numbers n.