Question

What is the sum of first n natural numbers ?

• A. $\frac{n(n+3)}{2}$
• B. $\frac{n(n+1)}{2}$
• C. $\frac{n^2(n+1)}{2}$
• D. $\frac{n(n^2+1)}{2}$

Hint:

The formula for the sum of the first $n$ natural numbers, $S_n = \frac{n(n+1)}{2}$, is a very common and useful result, often attributed to Carl Friedrich Gauss. It is derived from the properties of arithmetic progressions.

Answer 1

The Correct Option is B

Step 1: Understand what "natural numbers" mean.
Natural numbers are positive integers starting from 1: $1, 2, 3, \dots$ The sum of the first $n$ natural numbers means the sum $1 + 2 + 3 + \dots + n$. Step 2: Recall the formula for the sum of an arithmetic series.
The sum of an arithmetic series can be found using the formula:
$S_n = \frac{n}{2}(a_1 + a_n)$
where:
$S_n$ is the sum of the first $n$ terms
$n$ is the number of terms
$a_1$ is the first term
$a_n$ is the last term
Step 3: Apply the formula to the sum of natural numbers.
In the case of the sum of the first $n$ natural numbers:
$a_1 = 1$ (the first natural number)
$a_n = n$ (the $n$-th natural number)
$n = n$ (there are $n$ terms)
Substitute these values into the formula:
$S_n = \frac{n}{2}(1 + n)$
$S_n = \frac{n(n+1)}{2}$
Step 4: Compare the result with the given options.
The derived formula is $\frac{n(n+1)}{2}$, which matches option (2). (2) \(\frac{n(n+1)}{2}\)