# Prove by method of induction, for all n ∈ N: 2 + 4 + 6 + ..... + 2n = n (n+1) - Mathematics and Statistics

Sum

Prove by method of induction, for all n ∈ N:

2 + 4 + 6 + ..... + 2n = n (n+1)

#### Solution

Let P(n) ≡ 2 + 4 + 6 + …… + 2n = n(n + 1), for all n ∈ N

Step I:

Put n = 1

L.H.S. = 2

R.H.S. = 1(1 + 1) = 2 = L.H.S.

∴ P(n) is true for n = 1.

Step II:

Let us consider that P(n) is true for n = k

∴ 2 + 4 + 6 + ……. + 2k = k(k + 1) …(i)

Step III:

We have to prove that P(n) is true for n = k + 1 i.e., to prove that

2 + 4 + 6 + …. + 2(k + 1) = (k + 1) (k + 2)

L.H.S. = 2 + 4 + 6 + …… + 2 (k + 1)

= 2 + 4 + 6 + …… + 2k + 2(k + 1)

= k(k + 1) + 2(k + 1) …[From (i)]

= (k + 1).(k + 2)

= R.H.S.

∴ P(n) is true for n = k + 1

Step IV:

From all steps above by the principle of mathematical induction, P(n) is true for all n ∈ N.

∴ 2 + 4 + 6 + …… + 2n = n (n + 1) for all n ∈ N

Concept: Principle of Mathematical Induction
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#### APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board
Chapter 4 Methods of Induction and Binomial Theorem
Exercise 4.1 | Q 1 | Page 73