Question

Find the sum of even numbers between$1$ and$25$.

Answer 1

Step 1: Find the number of terms.

The even numbers between$1$ and $25$ are $2,4,6,7,8,\dots \dots \dots .24$

Here, the first term is, $a=2$

The common difference, $d=2$

$a_n=24$ (last term)

Now to find the number of terms i.e, $n$,

$$\begin{gathered} a_n=a+(n–1)d \\ \Rightarrow 24=2+(n–1)2 \\ \Rightarrow 24=2+2n–2 \\ \Rightarrow 2n=24 \\ \Rightarrow n=12 \end{gathered}$$

Step 2: Find the sum of all terms.

Applying the given data in the formula,

$S_n=\frac{n}{2}[2a+(n–1)d]$, we get

$$\begin{gathered} \Rightarrow S_{12}=\frac{12}{2}[2\times 2+(12–1)2] \\ \Rightarrow S_{12}=6[4+(11)2] \\ \Rightarrow S_{12}=6[4+22] \\ \Rightarrow S_{12}=6\left[26\right] \\ \Rightarrow S_{12}=156 \end{gathered}$$

Hence. the sum of even numbers between $1$ and $25$ is $156$.