# 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

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

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

 More topics in Series Formula Infinite Series Formula

#### Practise This Question

A horse is harnessed to a cart. If the horse tries to pull the cart, the horse must exert a force on the cart. By Newton's third law the cart must then exert an equal and opposite force on the horse. Since the two forces are equal and opposite, they must add to zero, so Newton's second law tells us that the acceleration of the system must be zero and therefore no matter how hard the horse pulls, it can never move the cart.