Prove that `2^(n) gt n` for all positive integers n.

Given that u_(n+1)=3u_n-2u_(n-1), and u_0=2 ,u_(1)=3 , then prove that u_n=2^(n)+1 for all positive integer of n

Find counter - examples to disprove the following statements : n^(2)-n+41 is a prime for all positive integers n.

Prove by induction that : 2^(n) gt n for all n in N .

Prove by induction that if n is a positive integer not divisible by 3 , then 3^(2n)+3^(n)+1 is divisible by 13 .

State and prove Bionomial theorem for any positive integer n.

If p is a natural number, then prove that p^(n+1) + (p+1)^(2n-1) is divisible by p^(2) + p +1 for every positive integer n.

Prove that n + n is divisible by 2 for any positive integer n.

Prove that (1+x)^(n) ge (1+nx) for all natural number n where x gt -1

If P(n):2^(n) lt n! Then the smallest positive integer for which P(n) is true if

Write the converse of the following statements. (i) If a number n is even, then n^(2) is even. (ii) If you do all the exercises in the book, you get an A grade in the class. (iii) If two integers a and b are such that a gt b , then a -b is always a positive integer.