NCERT solution Class 11 principle of mathematical induction

NCERT Solutions

Class 11 Maths

Principle of Mathematical Induction

NCERT Questions

Ex.4.1 Q.1
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Ex.4.1 Q.3
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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.18
Ex.4.1 Q.19
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Ex.4.1 Q.24

Ex.4.1 Q.18

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

1 + 2 + 3 + …………+ [n < (2n + 1)² ÷ 8]

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

P(n): 1 + 2 + 3 + …………+ [n < (2n + 1)² ÷ 8]

It can be noted that P(n) is true for n = 1 since

1 < (2.1 + 1)² ÷ 8 =

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

P(k): 1 + 2 + 3 + …………+ [k < (2k + 1)² ÷ 8] ………. (1)

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

Consider

1 + 2 + 3 + …………+ k + (k + 1) < {(2k + 1)² ÷ 8} + (k + 1)

[from equation 1]

1 + 2 + 3 + …………+ k + (k + 1) < {(2k + 1)² + 8(k + 1)} ÷ 8

1 + 2 + 3 + …………+ k + (k + 1) < {4k² + 4k + 1 + 8k + 8} ÷ 8

1 + 2 + 3 + …………+ k + (k + 1) < {4k² + 12k + 9} ÷ 8

1 + 2 + 3 + …………+ k + (k + 1) < (2k + 3)² ÷ 8

1 + 2 + 3 + …………+ k + (k + 1) < {2(k + 1) + 1}² ÷ 8

Thus, 1 + 2 + 3 + …………+ k + (k + 1) < (2k + 1)² ÷ 8 + (k + 1)

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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NCERT solutions of Principle of Mathematical Induction Class 11 Maths - all questions of all exercises are solved. Q. No. 1, 2, 3, and so on. Complete explanation and detailed solution makes them the best videos for Class 11 Maths Principle of Mathematical Induction NCERT solutions.