Question

Sigma Notation Write the sum using sigma notation.

$2+4+6+\mathrm{.....}+50$

Short answer

The sum using sigma notation is $\sum _{n=1}^{25}\left(2n\right)$.

Step-by-step solution

Step 1.  Given

The given sequence is $2+4+6+.....+50$.

Step 2.  To determine

We have to write the sum using sigma notation.

Step 3.  Calculation

We try to write the given sequence using a general term with n.

The given sequence is an arithmetic sequence with first term $a_1=2$and the common difference is $d=\text{second} \text{term} - \text{first} \text{term}=4-2=2$.

So, the nth term is:

$\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ =2+\left(n-1\right)\left(2\right)\\ =2+2n-2\\ =2n\end{array}$

Setting the last term equal to 2n we will solve for n.

$\begin{array}{c}2n=50\\ n=\frac{50}{2}\\ n=25\end{array}$

It means n goes from 1 to 25.

Hence, the sum is given by: $\sum _{n=1}^{25}\left(2n\right)$.