BAM FORM 5 – ALGEBRA

FINITE AND INFINITE SERIES

Sequence

Is a set of numbers in a defined order .

For example

2, 4, 6, 8, 10 ……………………..

3, 6, 9, 12, 15 …………………….

1, 4, 9, 16, 25……………………….

A term

Is the number in a sequence

For example

2, 4, 6, 8, 10,12…………

The 3^rd term is 6

The 6^th term is 12

Series

When the term of the sequence are joined by + or – sign we get the series

For example

2 + 4 + 6 + 8 +10 +…………………

3 + 6 + 9 + 12 +15 + ……………….

1+4 + 9 + 16 +25 +…………….

Finite series

Is the series which has an end

For example

1+ 4 +9 + 16

Infinite series

Is the series which does not have an end.

For example

1 +4 + 9+ 16+ ……………

Example

Find the term indicated against each of the following

-14, -10, -6 ……………. [6^th term]

2 + 4 + 6 …………….. [10^th term]

½ + ¼ + 1/8 + ………… [8^th term]

Solution

-14, – 10, -6……. … [6^th term]

The different by the way the consecutive is 4

        [-14+ 4] = -10

  [-10+4] = -6

 

The 6^th term is 6

1.              2+4+6+….. [ 10^th term ]

A_n = A₁ + [n-1] d

A₁₀= 2 + [10-1] 2

A ₁₀= 2 + [9 x 2]

A ₁₀ = 2+18

A ₁₀ = 20

The different by the conservative term is 2

2 + 4 +8+10+12+14+16+18+20

The 10^th term is 20

½ + ¼ + 1/8 …………… [8^th term ]

The series suggest that the nth term is

Therefore

The 8^th term is (½)⁸ = 1/ 256

SIGMA NOTATION (

Is the Greek letter used to indicate the sum of the series in a condensed form.

Example

The sum of the first term of the series

2+4+6+ ……….. Can be written as

2+4+6+8+10+12

2[1] + 2 [2] + 2 [3] + 2 [4] + 2 [5] + 2 [6]

In condensed form

,

       = 42

      Examples

1.   Write

              4[1] + 4 [2] + 4 [3] + 4 [4] +4 [5]

              4 +8+12+16+20

               = 60

               =3^1-2 + 3 ^1-3 + 3^1-4+ 3^1-5 +3^1-6

               =3^-1 + 3^-2 +3^-3+ 3^-4 +3^-5

               =1/3 + 1/32+ 1/33 + 1/34 + 1/35

               =1/3+1/9 +1/2 7 + 1/81+ 1/243

               = 

Find the sum of

      =[ 5+ [ 1-2] ]+ [ 5+ [ 2-2] ] + [ 5 + [ 3-2] ] [ 5 + [ 4-2] [ 5+[ 5-2] + [5+ [ 6-2]

      = [5 +-1] + [5+0] + [5+1] + [5+2] + [5+3] + [5+4]

     = 4+5+6+7+8+9

     = 39

Exercise

Find the term indicated against each of the following series

          a.)   3+6+9+………………. [ 7^th term ]

          b.)   1+4+9+……………… [ 11^th term ]

          c.)   1 + 1/3+ 1/5…………. [ 6^th term ]

          d.)   3+3+3+……………….. [ 21^st term ]

          e.)   1+1/2 +1/3…………….. [ 9^th term ]

Write the following series in expanded form

1).

2).

3).

Write the sum of the following

 .

 

 .

Write each of the following by using sigma notation

1. ½ +1/3 +1/4+1/5+1/6+1/7+1/8

2. ½ +2/3 +3/4+4/5+5/6+6/7

3. 3+9+27+81+243+729

4. 3+6+12+24+48

Solutions

 a) 3+6+9+……….

An = A₁ + [n-1] d

A₇ = 3+ [7-1] 3

A₇ =3+ [6×3]

A₇ = 3+18

A₇ = 21

The seventh term is 21

b) 1+4+9+……… [11^th term]

A_n = A₁+ [n-1] d

A ₁₁= 1+ [11-1] 5

A ₁₁= 1 + [10 x 5]

A ₁₁ = 1 +50

A ₁₁ = 51

The 11^th term is 51

c) 1+ 1/3 + 1/5+ …. [ 6^th term ]

A_n = A1 + [n-1] d

A ₆ = A1 + [6-1]

A₆= 1+5)

A ₆ = 1+

A ₆ = ()

d) A_n = A1+ [n-1] d

A ₁₁ = 1 + [11-1] 3

A ₁₁ = 1+ 10 [3]

A ₁₁ = 1+30

A ₁₁ = 31