The infinite sums of a sequence of numbers (or terms) is called a **series**. A sequence forms a pattern, these sequences can be identified through the patterns.

One important series (which happen to occur naturally in plants and nature) is Fibonacci series. The series go like this:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

As you can see, the number in any sequence is the sum of the two previous numbers - and it goes on forever!

A series follows a mathematical pattern and the terms can be described by a formula. We call any general or *n*th term of a series . where *n* stands for the number of the term and must be a positive integer.

For example, for the series 3 + 8 + 13 + 18 + 23 + ... find .

The first term is 3

Second term is 8 = 3 + 5 = 3 + 1 x 5

Third term is 13 = 3 + 5 + 5 = 3 + 2 x 5

Fourth term is 18 = 3 + 5 + 5 + 5 = 3 + 3 x 5

...and so on, following this pattern:

### Examples

1. The *n*th term of a series is given by . Which term of the series is equal to 104?

2. The *n*th term of a series is given by . Find the value of *n* for the first negative term in the series.

For the first negative term, we want:

Since *n* is an integer, *n*=34 i.e. the 34th term is the first negative term.

3. The *n*th term of a series is given by the formula . Which term of the series is equal to 4095?

## Sigma Notation

Series are often written in **sigma notation** () - which stands for the sum of a series. You may have seen this in finding the mean of a certain data in statistics.

### Examples

1. Evaluate

### Show Answer

means the sum of terms where the formula is with r starting at 1 and ending at 5.

So,

2. Write 7 + 11 + 15 + ... + (4k + 3) in sigma notation.

### Show Answer

When *n* = 1,

So the first term is at *n* = 1 and the last term is at *n = k*.

We can write the series as .

The number of terms in the series is *p - q + *1.