**Summary**

- When the terms of a sequence are added together a series is formed.
- A series can be denoted by a sigma notation where
*a*is the first value of*n*terms,*b*is the last value of the*n*terms and*x*is the expression of the given sequence. - The formula for
*n-th*term of an arithmetic progression is: - Sum of the series is given by the formula:
- The formula for
*n-th*term of a geometric progression is: - Formula to work out the sum of geometric progression:
- Sum to infinity of the geometric progression:

Let’s just quickly recall that sequence is a set of numbers in a given order with a rule for obtaining the terms. Moving on, when the terms of a sequence are added together a series is formed.

For example:

Sequence | Series |
---|---|

{1, 3, 5, 7….} | 1 + 3 + 5 + 7 + .... |

{5, 10, 15, 20….} | 5 + 10 + 15 + 20 + .... |

{10, 100, 1000,.....} | 10 + 100 + 1000 +..... |

A series can be denoted by a sigma notation where *a* is the first value of *n* terms, *b* is the last value of the *n* terms and* x* is the expression of the given sequence. This sigma notation is a Greek capital and is used to represent a sum.

Let’s consider we are given a series 2 + 4 + 6 + 8 + 10.

We can see that *n = 5* is the last value of n and the expression of the sequence would be .

Therefore, we can write it using the sigma notation in the form:

Next, to find the sum of the above sequence, we carry out the following steps:

Furthermore, in this article we will study series in the following types of sequences.

#### Arithmetic Progression

An arithmetic progression is a sequence of numbers which increases by a constant amount which could be either positive or negative. This amount is called the ”common difference” *(d)* and the starting number is called ”first term” *(a)*.

In general, if an arithmetic progression has first term *a* and common difference *d*, then in standard form, it is written as:

So the formula for* n-th* term of an arithmetic progression is:

Here: *a = first term*, *d = common difference*, *n = no. of terms*, *= n-th term*

#### Example #1

Q. Find the 20 th term of the sequence 5, 8, 11, 14, 17,…..

*Solution:*

We know a = 5, d = 8 – 5 = 3 also, we know it is an arithmetic sequence

** Ans**

#### Sum of Arithmetic Progression

Suppose we are asked to find the sum of first 100 numbers in an arithmetic progression. Adding all the numbers one by one will be a very tedious job. Gauss, a german mathematics solved this problem in a very simple manner.

If the first term is *a*, and the last term is *l*, the number of terms is *n*, then the sum of the series is given by the formula:

Or

#### Example #2

Q. Find the sum of the first 10 terms of the sequence 5, 8, 11, 14, 17,…..

*Solution:*

We know a = 5, d = 3 and n = 10. We know it is an arithmetic sequence.

*Ans*

#### Geometric Progression

Geometric progression is a sequence in which each term is multiplied by a constant *r* known as the common ratio. Thus if *a* is the first term, *r* is the common ratio, then the standard form of a geometric progression is:

Hence to find the term of the geometric progression, we use the formula:

Where *r* can be worked out by dividing two consecutive terms in such a way:

#### Example #3

Q. Find the 10th term of the geometric progression 3, 6, 12,….

*Solution:*

We know a = 3, n = 10 and we can find* r*

Use the above formula to work out the 10th term.

**Ans**

#### Sum of geometric progression

Just like we studied a formula to calculate the sum of arithmetic progression. Similarly, we are given a formula to work out the sum of geometric progression:

Here *a = first term*, r = common ratio, *n = no. of terms*

#### Sum to infinity of a geometric progression

Consider the following infinite geometric progression:

We can see that a = 1 and ,

As* n* increases decreases, when *n* becomes very large i.e , becomes very small and . The value of becomes .

We call the limiting value of as the **sum to infinity of the geometric progression.**

Therefore, if we consider -1 < r < 1, then when .

Hence the formula becomes:

#### Example #4

Q. Find the sum to infinity of the geometric progression:

Solution:

a = 2 and

Using the above formula:

**Ans**