# Series Formula

A series has a constant difference between terms. For example, 3 + 7 + 11 + 15 + ….. + 99. We name the first term as *a1*. The common difference is often named as “*d*”, and the number of terms in the series is *n. *

We can find out the sum of the arithmetic series by multiplying the number of times the average of the last and first terms.

The formula for finding out the sum of the terms of the arithmetic series is given as:

[\large x_{1}+x_{2}+x_{3}+….+x_{n}=\sum_{i-1}^{n}x_{i}]

[\large Sum=n\left(\frac{a_{1}+a_{n}}{2}\right)]

or

[\large \frac{n}{2}\left[2a_{1}+\left(n-1\right)d\right]]

### Solved Example

**Example:** 3 + 7 + 11 + 15 + ··· + 99 has *a1* = 3 and *d* = 4. Find *n*, using the explicit formula for an arithmetic sequence.

**Solution:**

We solve 3 + (*n* – 1) x 4 = 99 to get *n* = 25

$Sum=25\left(\frac{3+99}{2}\right)=1275$

$Sum=\frac{25}{2}\left[2\cdot 3+\left(25-1\right)\cdot 4\right]=1275$

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