__Sequence__ – An ordered list of numbers. (ie. 3, 6, 9, L2, 14, …… or 4,8, 16,32, ….). **Lorem Ipsum** is simply dummy text of the printing and typesetting industry. Lorem Ipsum has been the industry’s standard dummy text ever since the 1500s, when an unknown printer took a galley of type and scrambled it to make a type specimen book. It has survived not only five centuries, but also the leap into electronic typesetting, remaining essentially unchanged. It was popularised in the 1960s with the release of Letraset sheets containing Lorem Ipsum passages, and more recently with desktop publishing software like Aldus PageMaker including versions of Lorem Ipsum.

__Term__ – Each element of a sequence. The first term is referred to as u_{1} and the second term is u_{2} the third term is

u_{3}, and so on. The nth term is referred to as u_{n}.

Sequences may be *finite* (end at a certain term value) or *infinite* (continues to infinity) and denoted by three

dots at the end of the number sequence.

Sometimes a pattern can lead to a general rule for finding the terms of a sequence. The rule is called the

formula for the n^{th} term, u_{n}.

__Arithmetic Sequence__ – a sequence that has the same *common difference* between any pair of consecutive terms.

A common difference is obtained by subtracting a term from the previous term. lf this value is, constant for all terms in the sequence then the sequence is *arithmetic*.

i.e: 5,8, 11,14, 17, … common difference = 3

__General Term of an Arithmetic Sequence__

u_{n} = u_{1}+ (n – l)d

u_{n} = n^{th} term or general term

u_{n} = first term

n = number of terms

d = common difference

__Geometric Sequence__ – a sequence that has the same common ratio between any pair of consecutive terms. A common ratio is obtained by dividing any term by the preceding term. lf this value is

constant for all terms in the sequence then the sequence is *geometric*

i.e: 2,6,18, 54,…. common ratio = 3

__General Term of an Geometric Sequence__

u_{n} = u_{1}n^{n-1}

u_{n} = n^{th} term or general term

u_{n} = first term

n = number of terms

r = common ratio

__Sequences and summations__

The sum of the, first n terms of a sequence is

represented by summation notation.

S_{1000} = 1 + 2 + 3 + …..+ 999 + 1000

S_{1000} = 1000 + 999 + 998 + …..+ 1 + 2Example :

2S_{1000} = 1001 + 1001 + 1001 …..1001 + 1001 + 1001

2S_{1000 } = 1001(1001)

S_{1000 }= \(\frac{1001(1000)}{2}\)= 500500

__Example__: An empty barrel can hold L000 L of water. Alison pours 500 L of water into the barrel, then Jack pours 250 L into the barrel, then Marta pours 125 L into the barrel. Suppose the process goes on forever. Will the barrel ever fill? Will the barrel overflow?

__Solution__The amount may be written as a gyrometric series

500 + 250 + 125 ………..

The more term are add to the series. The close we get to 1000 L. we say limit of the infinite geometric series is 1000 L

__Sum of an Infinite Geometric Series__

S_{∞}= u _{1} + u _{1}r^{2} + u _{1}r^{3}……..

S_{∞}= \(\frac{u_1}{1-r}\) ,-1<r<1,(r<1)