Prove that if `rlt=slt=n ,t h e n ' n P_s` is divisible by `n P_r` .

If r lt s le n " then prove that " ^(n)P_(s) " is divisible by "^(n)P_(r).

Show that 2^(4n+4)-15n-16 ,w h e r en in N is divisible by 225.

If p is a fixed positive integer, prove by induction that p^(n +1) + (p + 1)^(2n - 1) is divisible by P^2+ p +1 for all n in N .

Prove that 2.7^(n)+ 3.5^(n)-5 is divisible by 24 for all n in N

Let f (n) = |{:(n,,n+1,,n+1),(.^(n)P_(n),,.^(n+1)P_(n+1),,.^(n+2)P_(n+2)),(.^(n)C_(n),,.^(n+1)C_(n+1),,.^(n+2)C_(n+2)):}| where the sysmbols have their usual neanings .then f(n) is divisible by

Let P(n) be the statement: " n^(3)+n is divisible by 3." i. Prove that P(3) is true. ii. Verify whether the statement is true for n = 4.

Statement 1: If p is a prime number (p!=2), then [(2+sqrt(5))^p]-2^(p+1) is always divisible by p(w h e r e[dot] denotes the greatest integer function). Statement 2: if n prime, then ^n C_1,^n C_2,^n C_2 ,^n C_(n-1) must be divisible by ndot

Prove that n^(2)-n divisible by 2 for every positive integer n.

Prove that a sequence in an A.P., if the sum of its n terms is of the form A n^2+B n ,w h e r eA ,B are constants.

If the sum of n terms of an A.P. is given by S_n=a+b n+c n^2, w h e r ea ,b ,c are independent of n ,t h e n a=0 common difference of A.P. must be 2b common difference of A.P. must be 2c first term of A.P. is b+c