# Using the princip

To Prove:

1 + 2 + 3 + 4 + … + n = 1/2 n(n + 1)

Steps to prove by mathematical induction:

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

(i) P(1) is true

(ii) P(k + 1) is true, whenever P(k) is true

Then P(n) is true for all n ϵ N

Therefore,

Let P(n): 1 + 2 + 3 + 4 + … + n = 1/2 n(n + 1)

Step 1:

P(1) = 1/2 1(1 + 1) = 1/2 × 2 = 1

Therefore, P(1) is true

Step 2:

Let P(k) is true Then,

P(k): 1 + 2 + 3 + 4 + … + k = 1/2 k(k + 1)

Now,

1 + 2 + 3 + 4 + … + k + (k + 1) = 1/2 k(k + 1) + (k + 1)

= (k + 1){ 1/2 k + 1}

= 1/2 (k + 1) (k + 2)

= P(k + 1)

Hence, P(k + 1) is true whenever P(k) is true

Hence, by the principle of mathematical induction, we have

1 + 2 + 3 + 4 + … + n = 1/2 n(n + 1) for all n ϵ N

Hence proved.

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