| 3. Use induction to prove that n² n is always even. -

1. Show that. (n + 1)(n + 2) n=1

Pravo by induction that no 2-1 for all positive intogors gors n.

pn+1 – 1 1. Use the formula 1+r +r² + .. + r" to evaluate the quantity r – 1 2+4+8+. + 4096 ... 2. Prove by induction that (п — 2)(п + 3) 3+4+5+... +n =

6. By Principle of Mathematical Induction prove 6n 6

3. Answer any one of 3(a) and 3(b). (a) Use mathematical induction to prove the following inequality: 4n > 1+ 3n for all positive integers n. (b) Prove that E, i (2²-1) = (n – 1)2" + 1 for all positive integers n using mathematical induction.

7. Use mathematical induction to show that п Σ 1 4i (4n+1 – 1) 3 i=0 for all integers n2 0.

(4a) Let n E Z. Prove that if n is even, then -5n – 3 is odd.

Let n 2 1, z be a real number, and r 2 -1. Prove the following statement using mathe- matical induction. (1 + r)" 2 1+ nx