Principle of Mathematical Induction – Class XI – Exercise 4.1[Part 2]

11: Prove the following by using the principle of mathematical induction for all n€ 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:

Let the given statement be P(n), i.e.,

P(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)

For n = 1, we have

P(1): 1/1.2.3 = 1.(1+3)/4(1+1)(1+2) = 1.4/4.2.3 = 1/1.2.3, which is true.

Let P(k) be true for some positive integer k, i.e.,

1/1.2.3 + 1/2.3.4 + 1/3.4.5 + … +1/k(k+1)(k+2) = k(k+3)/4(k+1)(k+2)

We shall now prove that P(k + 1) is true.

Consider Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.

1. Prove the following by using the principle of mathematical induction for all n€ N:

a + ar + ar2+ …+arn-1 = {a(rn– 1)} / {(r – 1)}

Solution:

Let the given statement be P(n), i.e.,

P(n): a + ar + ar2+ …+arn-1 = {a(rn– 1)} / {(r – 1)}

For n = 1, we have ,

P(1): a = {a(r1– 1)} / {(r – 1)}  = a which is true.

Let P(k) be true for some positive integer k, i.e.,

a + ar + ar2+ …+arn-1 = {a(rn– 1)} / {(r – 1)}  ————(1)

We shall now prove that P(k + 1) is true.

Consider Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.

13: Prove the following by using the principle of mathematical induction for all n € N:

(1+ 3/1) (1+ 5/4) (1+ 7/9)… (1+ (2n+1)/n2) = (n+1)2

Solution:

Let the given statement be P(n), i.e.,

P(n): (1+ 3/1) (1+ 5/4) (1+ 7/9)… (1+ (2n+1)/n2) = (n+1)2

For n = 1, we have

P(1): (1 + 3/1) = 4 = (1+1)2 = 22 = 4, which is true

Let P(k) be true for some positive integer k, i.e.,

(1+ 3/1) (1+ 5/4) (1+ 7/9)… (1+ (2k+1)/k2) = (k+1)2

We shall now prove that P(k + 1) is true.

Consider Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.

1. Prove the following by using the principle of mathematical induction for all n € N:

(1 + 1/1)(1 + 1/2)(1 + 1/3)…(1 + 1/n) = (n + 1)

Solution:

Let the given statement be P(n), i.e.,

P(n): (1 + 1/1)(1 + 1/2)(1 + 1/3)…(1 + 1/n) = (n + 1)

For n = 1, we have ,

P(1): (1 + 1/1) = 2 = (1+1) which is true.

Let P(k) be true for some positive integer k, i.e.,

P(k): (1 + 1/1)(1 + 1/2)(1 + 1/3)…(1 + 1/k) = (k + 1) ————-(1)

We shall now prove that P(k + 1) is true.

Consider Thus, P(k + 1) is true whenever P(k) is true. Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.

15: Prove the following by using the principle of mathematical induction for all n € N:

12 + 32 + 52 + … +(2n – 1)2 = n(2n-1)(2n+1)/3

Solution:

Let the given statement be P(n), i.e.,

P(n): 12 + 32 + 52 + … +(2n – 1)2 = n(2n-1)(2n+1)/3

For n = 1, we have

P(1) = 12 = 1 = 1(2.1-1)(2.1+1)/3 = 1.1.3/3 = 1, which is true.

Let P(k) be true for some positive ineteger k, i.e.,

P(k) = 12 + 32 + 52 + … +(2k – 1)2 = k(2k-1)(2k+1)/3 ———(1)

We shall now prove that P(k + 1) is true.

Consider Thus, P(k + 1) is true whenever P(k) is true. Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.

16: Prove the following by using the principle of mathematical induction for all n€ N:

1/1.4 + 1/4.7 + 1/7.10 + … + 1/(3n-2)(3n+1) = n/(3n+1)

Solution:

Let the given statement be P(n), i.e.,

P(n): 1/1.4 + 1/4.7 + 1/7.10 + … + 1/(3n-2)(3n+1) = n/(3n+1)

For n = 1, we have

P(1): 1/1.4 = 1/3.1+1 = 1/4 = 1/1.4, which is true.

Let P(k) be true for some positive integer k, i.e.,

P(k): 1/1.4 + 1/4.7 + 1/7.10 + … + 1/(3k-2)(3k+1) = k/(3k+1) ———-(1)

We shall now prove that P(k+1) is true.

Consider Thus, P(k + 1) is true whenever P(k) is true. Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.

1. Prove the following by using the principle of mathematical induction for all n € N:

1/3.5 + 1/5.7 + 1/7.9 + … + 1/(2n+1)(2n+3) = n/3(2n+3)

Solution:

Let the given statement be P(n), i.e.,

1/3.5 + 1/5.7 + 1/7.9 + … + 1/(2n+1)(2n+3) = n/3(2n+3)

For n = 1, we have ,

P(1): 1/3.5 = 1/3(2.1+3) = 1/3.5 which is true.

Let P(k) be true for some positive integer k, i.e.,

1/3.5 + 1/5.7 + 1/7.9 + … + 1/(2k+1)(2k+3) = k/3(2k+3) ————(1)

We shall now prove that P(k + 1) is true.

Consider Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.