## Sequence-II

In this page 'Sequence-II' we are going to see about sequences and nth term of the sequences.

We had seen some definition and examples in the page sequence.

We will see more about sequences here.

Sequences

Let us try this ......

2, 5, 8, 11, ...........

The next number is 14.  3 is added with the previous number.

So a sequence is an ordered list of things or elements.

In other words, sequences is a list of numbers or things that are arranged in particular pattern.

A sequence can be arranged in the following way.

a1,a2, a3, a4, .....an....

Here >a1,a2, a3, a4, an are called terms.

Let us see some more examples for sequences in this page 'Sequence-II'

1, 4, 9, 16, 25, ......

Here         a₁ =1 = 1²

a₂ = 4 = 2²

a₃ = 9 = 3²

a₄ = 16 = 4²   .....

The 10th term of the sequence is a₀ = 10² = 100 and the nth term of the sequence is n².

Some times we can easily find out the pattern. Some times it is hard to find out the pattern.

Now let us see another sequence.

1, 1, 2, 3, 5, 8, 13, ........

Let us find out the pattern here.

These numbers are called Fibonacci sequence named after the famous mathematician Fibonacci.

Formula to find the nth term of the sequences:

To find the nth term of the sequences we use the following formulas:

1. To find the nth term of an Arithmetic sequence:

If 'd' is the common difference between the two consecutive terms of AP,

d = t2 - t1
nth term of A.P

tn = a + (n-1)d

To find the nth term of Fibonacci sequence is

tn = tn-2 + tn-1.

We can build a sequence if the nth term is given.

2. To find the nth term of a sequence:

If the terms in a sequence go up by a same number for each term,  then the number will appear multiplied by n times in the formula for the nth term of the sequence.

Example:

Find the formula to find the nth term of the sequence

7, 11, 15, 19,....

Solution:  Here t₁ =  7

t₂ =  11

t₃ =   15

t₄  =  19   and so on.

The common difference is

t₄ - t₃  =    t₃ - t₂ =  t₂- t₁  = d

19 - 15 =  15 - 11 = 11- 7 = 4

So each term of this sequence go up by 4.

4n sequence :  4, 8, 12, 16, .....

The given sequence is

7, 11, 15, 19, ....

So to get the given sequence, let us add 3 for each term.

We had seen some examples in this page, 'Sequence-II' .  We will see more examples in the next page.