Sum of the first $n$ natural numbers, $1 + 2 + 3 + \ldots + n$, equals

Question

Accepted Answer

A. $n(n+1)/2$

1. This is an AP with first term $a = 1$, last term $l = n$, $n$ terms.
2. AP sum: $S_n = (n/2)(a + l) = (n/2)(1 + n) = \dfrac{n(n+1)}{2}$.
3. Examples: $1 + 2 + \ldots + 10 = 10 \cdot 11 / 2 = 55$ ✓; $1 + 2 + \ldots + 100 = 5050$ ✓.
4. Option B is approximate but wrong (missing the $/2$ for odd $n$ subtlety). Option C has no relation. Option D is the sum of SQUARES formula.
_Source: NCERT Class 11 Mathematics, Ch 8, §8.5 (Sum of first $n$ natural numbers), p. 12._

Answer

$n^2/2$

Answer

$n!/2$

Answer

$n(n+1)(2n+1)/6$