Problems on Mathematical Induction

Mathematical Induction For all n >= 1, prove that 12+22+32+42+.....+n2 = n(n+1)(2n+1)/6 For every positive integer n 13+23+33+43+.....+n3 = (n(n+1)/2 )2 For every positive integer n, prove that 7n-3n is divisible by 4. Show that (2.7n+3.5n-5) is divisible by 24. Prove that 102n-1+1 is divisible by 11 x2n-y2n is divisible by (x+y) 32n+2-8n-9 is divisible by 8 41n-14n is a multiple of 27 n(n+1)(n+5) is a multiple of 3 Using mathematical induction, prove that a+ar+ar2+ar3+ar4....arn-1 = a(rn-1)/(r-1)