Series and sequences

Series is the sum of the sequences.

Sequence2, 5, 8, 11, 14, 17, ................

Series :          2 + 5 + 8 + 11 + 14 + 17 + .......

We have special notation to denote series.

The nth term is denoted by an

.

We will use the Greek letter 'Σ' to denote the summation.

Σ  ←  Sigma

It starts with S which stands for summation.

How to denote series?

2 + 5 + 8 + 11 + 14 + 17

Here there are 6 termsLet us denote this in series notation. Σ an = a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 + a9 + a10

Any letter can be used for denoting index like 'i', 'j', 'k' and so on and 'n' used mostly.

Let us see some examples for series in this page 'Series and sequences.'

1. Let An = {1, 5, 9, 13, 17}.

What is the value of a3? Solution:

The index of the series is a₃ 3.

That is n=3.

That is the value of the third term.

The value of the third term is 9.

Next        a₁ + a₂ + a₃ + a₄ + a₅   =  1 + 5 + 9 + 13 + 17

=  45.

2.  Find out the nth term of the series 1 - 3 + 5 - 7 + 9 .....

Solution:

Here the terms are all odd numbers. So we can write the nth term as 2n+1 as it represent the odd number.

Here the alternate terms (that is even number of terms) are negative terms

When n= 1, the sign is positive

n =2, the sign is negative.

n = 3, the sign is positive and so on.

So the sign of the nth term is (-1)ⁿ⁺¹

The nth term is (-1)ⁿ⁺¹(2n+1) 3.  Write the 25th term of  5 + 9 + 13 + 17 + ....

Solution:

This is the summation of the sequence 5, 9, 13, 17, ......

So let us find the nth term the sequence 5, 9, 13, 17, ....

Here t₁ = 5

t₂ = 9

t₃ =  13

t₄ = 17, .....

Common difference : d  =  t₄ - t₃ = t₃ - t₂ = t₂ - t

17 - 13 = 13 - 9 = 9 - 5

4     =      4     =  4

So each term is increased by 4. So 4n will come in the formula.

4n sequence :  4, 8, 12, 16, 20, .......

Given sequence: 5, 9, 13, 17, ....

So the nth term of the sequence is 4n + 1.

Here t25 = 4(25) + 1

t₂₅ = 100 + 1  =  101.

We had seen some examples for series in this page 'Series and sequences.' We will see more examples in the following pages.