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Chapter 12: Q. 1 (page 830)

use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n.

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

Short Answer

Expert verified

We proves that the statement is true using the mathematical induction

Step by step solution

01

Given information

We are given a sequence

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

02

We check for n=1

LHS= 2

RHS=1(1+1)

=2

hence LHS=RHS

Hence the statement is true for n=1

03

Assume the statement to be true for n

We get,

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

04

Now we prove that the statement is true for n+1

That is we have to prove that

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

Consider the left hand side

We have,

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

Hence proved the statement is true for all n belonging to natural numbers

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