**TOPIC 5: SEQUENCE AND SERIES ~ MATHEMATICS FORM 3**

Sequences

The Concept of Sequence

Explain the concept of sequence

A Sequence is the arrangement of numbers or is a list of numbers following a clear pattern such that one number and the next are separated by comma (,).

Example: a

_{1}, a_{2}, a_{3}, a_{4 }……………………..NB: Each number found in a Series or Sequence is called a

**.***term*Example 1

Find the next three terms in the following sequences.

- 5, 8, 11, 14, 17,………………………………
- 3, 7, 6, 10, 9, …………………………………
- 1, 2, 4, 7, ………………………………………
- 2, 9, 20, 35, …………………………………

*Solution:*(a)You can see that each term is less to the next by 3.

So next three terms are (17+3),(17+3+3) and 17+3+3×3)

Which are 20, 23, and 26

Alternately add 4 and subs tract 1. The sequence then extends to 13, 12, 16

We see that the difference is increasing by 1 each time. So the next three terms are 11, 16 and 22.

The differences are increased by 4 each time, so the next three terms are 54, 77 and 104.

Example 2

Write down the first three terms in the sequences where the n

^{th}term is given by the formulae.Example 3

The k

^{th}term of a series is k^{2 }+ 4Find the sum of the first four terms in the series

*Solution**:*

k=1, k

^{2}+4=1^{2}+4=5k=2, k

^{2}+4=2^{2}+4=8k=3, k

^{2}+4=3^{2}+4=13k=4, k

^{2}+4=4^{2}+4=20So the series is 5+8+13+20 and its sum is

**46**Example 4

Find the n

^{th}term of the following sequences:Exercise 1

1. Write down the next three terms in the following sequences

2. Find the first three terms in the sequence:

- 5n+2
- 1-3k
- n
^{2}+n+1 - 2
^{n}

3. Find the sum of the first four terms of the series where the k

^{th}term is given by:- 5k+3
- k
^{3}-1 - 2
^{k}

4. Find the n

^{th}term of these sequences:An Arithmetic Progression (AP) and Geometric Progression (GP)

Identify an arithmetic progression (AP) and geometric progression (GP)

When

the series or sequence is such that between two consecutive terms there

is a difference which is fixed, then the series or sequence is called

an arithmetic progression (A.P)

the series or sequence is such that between two consecutive terms there

is a difference which is fixed, then the series or sequence is called

an arithmetic progression (A.P)

The fixed difference (number) between two consecutive terms is called the common difference (d)

Example 5

In the sequence 4, 7, 19, 13, 16 there is a common difference which is

7-4=10-7=13-10=16-13=3.

So the common difference (d)=3.

Note that in arithmetic progression (A.P) the difference between two successive terms is always the same.

Sometimes

numbers may be decreasing instead of increasing, the arithmetic

sequence or series while terms decrease have a negative number as a

common difference.

numbers may be decreasing instead of increasing, the arithmetic

sequence or series while terms decrease have a negative number as a

common difference.

Example 6

The common difference of the sequence 6, 4, 0, -2, …………………… is

4-6=2-4=0-2=-2-0=-2

So the common difference is –

**2.**In general if A

_{1}, A_{2}, A_{3}, A_{4}, ……………………… A_{n}are the terms of the arithmetic sequence , then the common difference is ;Example 7

For each of the following sequences, find the common difference and write the next two terms.

Solution:

Exercise 2

1. Find the common difference for each of the following sequence:

- 11, 14, 17, 20, …………………………………
- 2, 4, 6, 8, 10, ……………………………………
- 0.1, 0.11, 0.111, 0.1111 , …… … … … … …
- y, y+3, y+6, y+9, y+12, … …… … … … ……

2. State whetherthe following sequence are arithmetic or not:

- 2, 5, 8, 11, 14, …………… ……………… ……
- 1, 3, 4, 6, 7, 9, 10, ………………………………
- y, y + x, y+2x, y+3x, … ………… ……

3. The temperatureat a mid day is 3

^{0}c, and it falls by 2^{0}c each hour. Find the temperature at the end of the next four hours.Geometric Progression (G.P).

When

the series or Sequence is such that between two consecutive terms there

is a ration which is fixed, then the series or sequence is called a

geometric progression (G.P)

the series or Sequence is such that between two consecutive terms there

is a ration which is fixed, then the series or sequence is called a

geometric progression (G.P)

The fixed ratio(number) between two successive terms is called the common ratio (r).

Example 8

In 2, 4, 8, 16, 32, … … …… … … …….

There is a common ration which is

Note that like in arithmetic progression (A.P), in geometric progression (G.P) the common ratio does not change.

Also

the terms may be decreasing instead of increasing, the geometric

sequence or series whose terms decrease have a positive common ratio

which is less than 1 for the progression with positive terms.

the terms may be decreasing instead of increasing, the geometric

sequence or series whose terms decrease have a positive common ratio

which is less than 1 for the progression with positive terms.

Example 9

For each of the following sequence find the common ratio.

Example 10

For the following geometric sequences, find the common ratio and write down the next two terms:

The next term is found by multiplying the term considered to be the last term by the common ratio.

Exercise 3

1. Which of the following sequences are geometric

- 1, 2, 4, 8, 16, ……………………………………
- 2, 6, 18, 54, 162, …………………………………
- 1, -1,1,-1,1, ………………………………………
- x
^{2}, 2x^{3}, 4×4, 8x^{3}………………………………… - 1, 2, 4, 7, 10, ………………………………………
- 0.1, 0.2, 0.3, 0.4, 0.5, ……………………………
- 3, 6, 9, 12,15, ……………………………………….

2. Find thecommon difference for each of the following geometric progressions (G.P)

3. Find thenext term of the sequence 2, 10, 50, 500,………………….

4.

The populationof a town is decreasing so that every year the population

declines by a quarter. If the population is originally 100,000. What

will it be after 5 years?

The populationof a town is decreasing so that every year the population

declines by a quarter. If the population is originally 100,000. What

will it be after 5 years?

The General Term of an AP

Find the general term of an AP

If A

_{1}, A_{2}, A_{3}, …………………A_{n }are the terms of an arithmetic sequence, then there is a common difference d which is given byd = A

_{2}– A_{1}= A_{3}– A_{2}= A_{n}– A_{n}– 1But . A

_{3 = }A_{1}+ 2d which meansA

_{4 }=[ A_{1}+2d]+d= A

_{1}+ 3dPutting into consideration this pattern, it is true that

A

_{5 }= A_{1}+ 4dA

_{6 }= A_{1}+ 5dA

_{n}= A_{1}+ (n-1)dWhere A

_{n}is the n^{th}termThe n

^{th}term of the sequence with first term A_{1 }and common difference d is given byExample 11

Find theformula for the n

^{th}term of the sequence 8 , 9.5, 11, 12.5, 14, 15.5,……Note that the n

^{th}term gives every term in the sequence,For example when n=3, you have A

_{3}=1.5×3+6.5=11So A

_{3}=11 where 11 is given in the sequence above having the third position.Therefore A

_{n}shows the position of the term in sequence and of A_{1}+(n-1)d gives the value of the term for any positive integer.Example 12

The 5

^{th}term of an arithmetic sequence is 11, and the 8^{th}term is 26. Find the first five terms.Example 13

The 8

^{th}term of an arithmetic sequence is 9 greater than the 5^{th}term, and the 10^{th}term is 10 times the 2^{nd}term. Find- The common difference (d)
- 20
^{th}term.

The General Term of GP

Find the general term of GP

If G

_{1}, G_{2}, G_{3},……………..G_{n}are the terms of a geometric sequence, then they have a common ratio (r) which is given byExample 14

Find the formula for the n

^{th}term of each of the following geometric sequence.- 2, 6, 18, 54 , ………………………………
- 4,-2, 1, -0.5, 0.25 …………………………

Example 15

Considering that,

Exercise 4

1. In the arithmetic sequence, the 17

^{th}term is 30 and 9^{th}term is 42 find the first three terms.2. In the Arithmeticsequence the third term 12 and the 9

^{th}term 24. Find the n^{th}term of the sequence and use it to find the 15^{th}term.3. Find the15

^{th}term of the sequence 5, 10, 20, 40 ,…………………………4.

A population isincreasing and every year it is multiplied by 1.03. If

it starts off at 10,000,000, what will it be after n years?

A population isincreasing and every year it is multiplied by 1.03. If

it starts off at 10,000,000, what will it be after n years?

5. The first termof the geometric sequence is 7 and the common ratio is 4. What is the 9

^{th}term of this sequence?
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