Question

Prove the following by induction: 2 + 4 + 6 + …+ 2n = n(n+1) for all integers n

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Answer 1

We have to prove the following statement using mathematical induction:

BASIS STEP

We first check if the statement holds true for , clearly

So, the statement holds true for

INDUCTION HYPOTHESIS

Now, we assume that the statement holds true for , that is,

............................................(1)

INDUCTION STEP

Now, we use the earlier assumption as check whether the statement holds true when

We have

Using (1) in the above expression we have

Taking (k+1) common we get

which is in the form n(n+1). Hence this shows that the statement holds true for whenever it holds true for , hence, by the principle of mathematical induction, the statement must hold true for all natural numbers. Thus, we have