Question

Find the sum of all even numbers from 1 to 350.

Answer 1

The given sequence is 2, 4, 6, 8 ... 350
First term (a) = 2
Common difference (d) = 2

Now,
$$\begin{gathered} a_n=a+\left(n-1\right)d \\ \Rightarrow 350=2+\left(n-1\right)2 \\ \Rightarrow 350=2+2n-2 \\ \Rightarrow 2n=350 \\ \Rightarrow n=175 \end{gathered}$$

Thus, the number of terms (n) = 175.

We know that,
$$\begin{gathered} S_n=\frac{n}{2}\left(2a+\left(n-1\right)d\right) \\ \therefore S_{175}=\frac{175}{2}\left(2\left(2\right)+\left(175-1\right)\left(2\right)\right) \\ =\frac{175}{2}\left(4+174\left(2\right)\right) \\ =\frac{175}{2}\left(4+348\right) \\ =\frac{175}{2}\times 352 \\ =30800 \end{gathered}$$

Hence, the sum of all even numbers from 1 to 350 is 30800.