Page 11 - 11-Math-8 Mathematical Inductions and Binomial Theorem
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19. 12 + 32 + 52 + .... + (2n -1)2 =n(4n2 -1)
3
20.  3  +  4  +  5  + .... +  n+ 2  =ï£ï£¬ï£« n4+ 3
 3   3   3   3  
ï£ ï£¸ ï£ ï£¸ ï£ ï£¸ ï£ ï£¸ 
21. Prove by mathematical induction that for all positive integral values of n
i) n2 + n is divisible by 2. ii) 5n - 2†is divisible by 3.
iii) 5n -1 is divisible by 4. iv) 8 x 10n - 2 is divisible by 6.
v) n3- n is divisible by 6.
22. 1 + 1 + .... + 1 = 1 1 - 1 
3 32 3n 2 3n 
23. 12 - 22 + 32 - 42 + .... + (-1)n-1 .n2 =(-1)n-1.n(n + 1)
2
24. 13 + 33 + 53 + .... + (2n -1=)3 n2[2n2 -1]
25. x + 1 is a factor of x2n -1;(x ≠-1)
26. x - y is a factor of xn - yn ; (x ≠y)
27. x + y is a factor of x2n-1+ y2n-1 (x ≠-y )
28. Use mathematical induction to show that
1 + 2 + 22 +.... + 2n = 2n+1 - 1 for all non-negative integers n.
29. If A and B are square matrices and AB = BA, then show by mathematical induction that
ABn = BnA for any positive integer n.
30. Prove by the Principle of mathematical induction that n2 - 1 is divisible by 8 when n is
an odd positive integer.
31. Use the principle of mathematical induction to prove that lnxn = n lnx for any integer
n 8 0 if x is a positive number. Use the principle of extended mathematical induction
to prove that:
32. n! > 2n - 1 for integral values of n 8 4 .
33. n2 > n + 3 for integral values of n 8 3.
34. 4n > 3n + 2n-1 for integral values of n 8 2 .
35. 3n < n! for integral values of n > 6 .
36. n! > n2 for integral values of n 8 4 .
version: 1.1
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