(1) By the principle of mathematical induction, prove that, for n ≥ 1
13 + 23 + 33 + · · · + n3 = [n(n + 1)/2]2
(2) By the principle of mathematical induction, prove that, for n ≥ 1
12 + 32 + 52 + · · · + (2n − 1)2 = n(2n − 1)(2n + 1)/3
(3) Prove that the sum of the first n non-zero even numbers is n2 + n. Solution
(4) By the principle of mathematical induction, prove that, for n ≥ 1
1.2 + 2.3 + 3.4 + · · · + n.(n + 1) = n(n + 1)(n + 2)/3
(5) Using the Mathematical induction, show that for any natural number n ≥ 2,
(1 − 1/22) (1 − 1/32)(1 − 1/42) ...............(1 − 1/n2) =(n + 1)/2n
(6) Using the Mathematical induction, show that for any natural number n ≥ 2,
[1/(1 + 2)] + [1/(1 + 2 + 3)] + [1/(1 + 2 + 3 + 4)] + · · · + [1/(1 + 2 + 3 + · · · + n)] = (n − 1)/(n + 1) Solution
(7) Using the Mathematical induction, show that for any natural number n,
[1/(1.2.3)]+[1/(2.3.4)]+[1/(3.4.5)]+ · · · +[1/(n.(n + 1).(n + 2))]
= n(n + 3)/4(n + 1)(n + 2) Solution
(8) Using the Mathematical induction, show that for any natural number n,
1/(2.5) + 1/(5.8) + 1/(8.11) + · · · + 1/(3n − 1)(3n + 2) = n/(6n + 4) Solution
(9) Prove by Mathematical Induction that
1! + (2 × 2!) + (3 × 3!) + ... + (n × n!) = (n + 1)! − 1.
(10) Using the Mathematical induction, show that for any natural number n, x2n − y2n is divisible by x + y. Solution
(11) By the principle of Mathematical induction, prove that, for n ≥ 1, 12 + 22 + 32 + · · · + n2 > n3/3 Solution
(12) Use induction to prove that n3 − 7n + 3, is divisible by 3, for all natural numbers n. Solution
(13) Use induction to prove that 10n + 3 × 4n+2 + 5, is divisible by 9, for all natural numbers n. Solution
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