**Mathematical Induction Worksheet With Answers :**

Here we are going to see some mathematical induction problems with solutions.

**Define mathematical induction :**

Mathematical Induction is a method or technique of proving mathematical results or theorems

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

1^{3} + 2^{3} + 3^{3} + · · · + n^{3} = [n(n + 1)/2]^{2}

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

1^{2} + 3^{2} + 5^{2} + · · · + (2n − 1)^{2} = n(2n − 1)(2n + 1)/3

(3) Prove that the sum of the first n non-zero even numbers is n^{2} + 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 − 1/2^{2}) (1 − 1/3^{2})(1 − 1/4^{2}) ...............(1 − 1/n^{2}) =(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, x^{2n} − y^{2n} is divisible by x + y. Solution

(11) By the principle of Mathematical induction, prove that, for n ≥ 1, 1^{2} + 2^{2} + 3^{2} + · · · + n^{2} > n^{3}/3 Solution

(12) Use induction to prove that n^{3} − 7n + 3, is divisible by 3, for all natural numbers n. Solution

(13) Use induction to prove that 10^{n} + 3 × 4^{n+2} + 5, is divisible by 9, for all natural numbers n. Solution

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