1. For each n $\in N,2^{3n}-1$is divisible by

2. x ($x^{n-1}-n{a}^{n-1})+a^n(n-1)$is divisible by (x$-a{)}^2$ for

3. $2^{3n}-7n-1$is divisible by

4. If n is a positive integer, then $n^3+2n$is divisible by

5. If m, n are any two odd positive integer with n $<$ m, then the largest positive integers which divides all the numbers of the type $m^2-$ $n^2$ is

6. The smallest positive integer for which the statement $3^{n+1}$$<$ $4^n$ holds is

7. P(n) : $2^{n+2}$ $<$ $3^n$, is true for

8. If P(n) : $3^n$ $>$ 4n, then P(n) is true for

9. The inequality n!$>$$2^{n-1}$ is true

10. P(n):$3^{2n+2}-8n-9$ is divisible by 64, is true for

11. The statement P(n):1$\times 1!+2\times 2!+3\times 3!+\mathrm{....}+n\times n!=(n+1)!-1$ is

12. A student was asked to prove a statement by induction. He proved (i) P(5) is true and (ii) truth of P(n)$\Rightarrow$ truth of P(n+), n $\in N$. On the basis of this, he could conclude that P(n) is true

13. The greatest positive integer which divides (n+1)(n+2)(n+3)......(n+r)for n$\in N$is

14. Let P(n): $n^2+n$is an odd integer. It is seen that truth of P(n) $\Rightarrow$the truth of P(n+1). Therefore P(n) is true for all

15. The greatest positive integer, which divides (n+16)(n+17)(n+18)(n+19), for all n $\in N,$is

16. If P(x): 2+4+6+....+2n, n$\in N,$then P(m) =m(m+1)+2
$\Rightarrow P(m+1)=(m+1)(m+3)+2\forall m\in N\mathrm{.}$ So we can conclude that P(n) = n(n+1)+2 for

17. The smallest +ve integer n for which n!$<(\frac{n+1}{2}{)}^n$holds is

18. If X $>-1,$then the statement P(n): $(1+x{)}^n>1$+ nx is true for

19. If P(n): 2n $<n!,n\in N,$then P(n) is true for

20. If P(n) is a statement such that truth of P(n) $\Rightarrow$ the truth of P(n+1) for n $\in N,$then P(n) is true