Question

Prove that (

1

2−2

+

1

2−2

) 22 =

1−22

2+22​

Answer 1

Step-by-step explanation:

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Answer 2

Answer:

4.1 Overview

Mathematical induction is one of the techniques which can be used to prove variety

of mathematical statements which are formulated in terms of n, where n is a

positive integer.

4.1.1 The principle of mathematical induction

Let P(n) be a given statement involving the natural number n such that

(i) The statement is true for n = 1, i.e., P(1) is true (or true for any fixed natural

number) and

(ii) If the statement is true for n = k (where k is a particular but arbitrary natural

number), then the statement is also true for n = k + 1, i.e, truth of P(k) implies

the truth of P(k + 1). Then P(n) is true for all natural numbers n.

4.2 Solved Examples

Short AnswerType

Prove statements in Examples 1 to 5, by using the Principle of Mathematical Induction

for all n ∈ N, that :

Example 1 1 + 3 + 5 + ... + (2n – 1) = n

2

Solution Let the given statement P(n) be defined as P(n) : 1 + 3 + 5 +...+ (2n – 1) =

n

2

, for n ∈ N. Note that P(1) is true, since

P(1) : 1 = 12

Assume that P(k) is true for some k ∈ N, i.e.,

P(k) : 1 + 3 + 5 + ... + (2k – 1) = k

2

Now, to prove that P(k + 1) is true, we have

1 + 3 + 5 + ... + (2k – 1) + (2k + 1)

= k

2

+ (2k + 1) (Why?)

= k

2

+ 2k + 1 = (k + 1)

Step-by-step explanation:

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