NCERT solution Class 11 principle of mathematical induction

NCERT Solutions

Class 11 Maths

Principle of Mathematical Induction

NCERT Questions

Ex.4.1 Q.1
Ex.4.1 Q.2
Ex.4.1 Q.3
Ex.4.1 Q.4
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Ex.4.1 Q.7
Ex.4.1 Q.8
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Ex.4.1 Q.11
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Ex.4.1 Q.15
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Ex.4.1 Q.19
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Ex.4.1 Q.21
Ex.4.1 Q.22
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Ex.4.1 Q.24

Ex.4.1 Q.5

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

1.3 + 2.3² + 3.3³ + n3ⁿ = {(2n - 1)3^n-1 + 3} ÷ 4

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

P(n): 1.3 + 2.3² + 3.3³ + n3ⁿ = {(2n - 1)3^n-1 + 3} ÷ 4

For n = 1, we have

P (1): 1.3 = 3 = {(2 × 1 - 1)3^1-1 + 3} ÷ 4 = (3² + 3) ÷ 4 = 12 ÷ 4 = 3, which is true.

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

1.3 + 2.3² + 3.3³ + k3^k = {(2k - 1)3^k-1 + 3} ÷ 4 ………… (1)

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

Consider

1.3 + 2.3² + 3.3³ + ... + k.3^k + (k + 1).3^k+1

= (1.3 + 2.3² + 3.3³ + ... + k.3^k) + (k + 1).3^k+1

= [{(2k - 1)3^k-1 + 3} ÷ 4] + (k + 1).3^k+1

[Using equation 1]

= {(2k - 1)3^k-1 + 3 + 4(k + 1).3^k+1} ÷ 4

= [3^k+1{2k - 1 + 4(k + 1)} + 3] ÷ 4

= {3^k+1(6k + 3) + 3} ÷ 4

= {3^k+1× 3(2k + 1) + 3} ÷ 4

= {3^{k+1+ 1}(2k + 1) + 3} ÷ 4

= [3^{(k+1) + 1} {2(k + 1) – 1} + 3] ÷ 4

= [2(k + 1) – 1}3^{(k+1) + 1} + 3] ÷ 4

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.

Principle of Mathematical Induction Class 11 Maths NCERT Chapter 4 NCERT Solutions

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