Question

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

We proves that the statement is true using the mathematical induction

Step-by-step solution

Given information

We are given a sequence

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

We check for n=1

LHS= 2

RHS=1(1+1)

=2

hence LHS=RHS

Hence the statement is true for n=1

Assume the statement to be true for n

We get,

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

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,

$$\begin{gathered} 2+4+6+....+2n+2(n+1) \\ =n(n+1)+2(n+1) \\ =(n+1)(n+2) \\ =RHS \end{gathered}$$

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