1. For all n ∈ N, 3×5²ⁿ⁺¹ + 2³ⁿ⁺¹ is divisible by

2.  If P(n) : (2n + 7) < (n + 3)² then P(3) is

3. (1 + x)n ≥ ____ for all n ∈ N,where x > -1

4. The sum of cubes of three consecutive natural numbers is divisible by:

5. A set S in which x S implies x+1 S is known as a _______ .

6. If a statement is to be proved by mathematical induction, then the different steps necessary to prove it are

7. A student was asked to prove a statement P (n) by method of induction. He proved that P (3) is true and that P(n) ⇒ P(n+1) for all natural numbers n. On the basis of this he could conclude that P (n) is true

8. For any natural number n, 7ⁿ – 2ⁿ is divisible by

9. The greatest positive integer, which divides n (n + 1) (n + 2) (n + 3) for all n ∈ N, is

10. Find the number of shots arranged in a complete pyramid the base of which is an equilateral triangle, each side containing n shots.

11. For all n∈N, 7²ⁿ − 48n−1 is divisible by :

12. The sum of the series 1² + 2² + 3² + ………..n² is

13. If n is an odd positive integer, then aⁿ + bⁿ is divisible by :

14. For all n ∈ N , 49ⁿ+16n−1 is divisible by

15. (n² + n) is ____ for all n ∈ N.

16. {1 – (1/2)}{1 – (1/3)}{1 – (1/4)} ……. {1 – 1/(n + 1)} =

17. {1/(3 ∙ 5)} + {1/(5 ∙ 7)} + {1/(7 ∙ 9)} + ……. + 1/{(2n + 1)(2n + 3)} =

18. If n is a positive integer , then 2.7ⁿ+3.5ⁿ−5 is divisible by

19. Let P(n) b e a statement 2ⁿ<n! where n is a natural number, then P(n) is true for

20. 10^2n-1 + 1 is divisible by ____ for all N ∈ N