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What is the sum of the first $100$ natural numbers?

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- 5050

5005

5050

- 5005

answer is C.

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Detailed Solution

It is given the natural numbers 1,2,3……100.
The natural number form an AP with first term 1 and common difference 1.
So, we have,

a = 1, d = 1

.
We know that sum of first n terms of an AP is given as,

S_{n} = \frac{n}{2} [2 a + (n - 1) d]

.
Putting the value of a = 1 and d = 1, we get

\Rightarrow S_{n}

= \frac{n}{2} [2 \times 1 + (n - 1) \times 1]

\Rightarrow S_{n} = \frac{n}{2} [2 + n - 1]

\Rightarrow S_{n} = \frac{n}{2} [n + 1]

\Rightarrow S_{n} = \frac{n (n + 1)}{2}

[1]
Substituting

n = 100

in [1], we get

S_{100} = \frac{100 (100 + 1)}{2}

{\Rightarrow S}_{100} = 50 \times 101

{\Rightarrow S}_{100} = 5050

Therefore, the sum of the first 100 natural numbers is 5050.
Hence, the correct option is 3.

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