The sum of the first \( n \) natural numbers is given by: \( \displaystyle S_n = \frac{n(n + 1)}{2} \). In this post, we will learn how to prove it.
Proof:
We will prove it using mathematical induction:
- Base Case \( ( n = 1 ) \):
\( \displaystyle S_1 = 1 = \frac{1(1 + 1)}{2} = 1 \)
The formula holds for \( n = 1 \).
- Inductive Step:
Assume the formula holds for \( n = k \), i.e.,
\( \displaystyle S_k = 1 + 2 + \dots + k = \frac{k(k + 1)}{2} \)
Prove for \( n = k + 1 \):
\( \begin{aligned} &S_{k+1} = S_k + (k + 1) \\ &= \frac{k(k + 1)}{2} + (k + 1) \\ &= \frac{k(k + 1) + 2(k + 1)}{2} \\ &= \frac{(k + 1)(k + 2)}{2} \\ \end{aligned} \)Thus, the formula holds for \( n = k + 1 \).
Thus, by induction, the formula is true for all \( n \in \mathbb{N} \).
Please let me know in the comments if you find any errors in this proof.