2) Prove that 1 + 3n < 4n for all n > 1. /5 Marks/
2. Prove that if n > 1, then 1(1!) + 2(2!) + ... + n(n!) = (n + 1)! - 1.
.n= n(n-1)(n+1) for all n > 2. 12. Use induction to prove (1 : 2) +(2-3)+(3-4) +...+(n-1).n [9 points) 3
Prove using mathematical induction that 3" + 4" < 5" for all n > 2.
2. (D5) Let n = o(a) and assume that a =bk. Prove that <a >=<b> if and only if n and k are relatively prime.
(1) Prove that for Res > 3 (s)S(s - 2)- n-1 where σ2(n)-2.1n d2 is the sum of the squares of the positive divisors of n.
5. Use Rice's Theorem to prove the undecidablity of the following language. P = {< M > M is a TM and 1011 E L(M)}.
| Prove that for n e N, n > 0, we have 1 x 1!+ 2 x 2!+... tnx n! = (n + 1)! - 1.
Problem 44) Prove: n!> 2" for n24. Problem 45) Prove by induction: For n>0·AT- i=1
Use the Principle of Mathematical Induction to prove that (2i+3) = n(n + 4) for all n > 1.