MATHEMATICAL INDUCTION WORKSHEET WITH ANSWERS

(1)  By the principle of mathematical induction, prove that, for n ≥ 1

13 + 23 + 33 + · · · + n3 = [n(n + 1)/2]2

Solution

(2)  By the principle of mathematical induction, prove that, for n ≥ 1

12 + 32 + 52 + · · · + (2n − 1)2 = n(2n − 1)(2n + 1)/3

Solution

(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

Solution

(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

Solution

(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.

Solution

(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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