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

Show that ""^(n)P_(n)=""^(n)P_(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)

prove that (1-(1)/(2^(2)))(1-(1)/(3^(2)))(1-(1)/(4^(2)))(1-(1)/(n^(2)))=(n+1)/(2n) for all natural numbers,n>=1(1-(1)/(n^(2)))=(n+1)/(2n)

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

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

A sequence b_(0),b_(1),b_(2), . . . is defined by letting b_(0)=5 and b_(k)=4+b_(k-1) , for all natural number k. Show that b_(n)=5+4n , for all natural number n using mathematical induction.

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