Sequences and Series (4)Sequences and Series (4)

Learn what is meant by an Learn what is meant by an

ARITHMETIC SERIESARITHMETIC SERIES

The can pyramid…The can pyramid…

How many cans are there in this pyramid.

How many cans are there in a pyramid with 100 cans on the bottom row?

Arithmetic SeriesArithmetic Series

An Arithmetic series is the sum of the terms in an Arithmetic sequence.

Eg. 1, 2, 3, 4… (Arithmetic sequence)

1 + 2 + 3 + 4… (Arithmetic series)

Back to the pyramid…Back to the pyramid…

We wanted to work out the sum of:

1 + 2 + 3 + ….. + 98 + 99 + 100

100 + 99 + 98 + ….. + 3 + 2 + 1If we write it out in reverse we get….

101 + 101 + 101 +…..

How many times do we add 101 together?101 x 100 = 10100

What do we need to do to this answer?10100 / 2 = 5050

Activity 1Activity 1

Work out the sum of the first 50 positive integers.

Work out the sum of all the odd numbers from 21 up to 99.

Arithmetic SeriesArithmetic SeriesWork out the sum of all the odd numbers from 21 up to 99.

a (a + d)+ + (a + 2d) + (a + 3d) (l - 2d)+ + (l - d) + l(l - 3d)+ …. +

l (l - d)+ + (l - 2d) + (l - 3d) (a + 2d)+ + (a + d) + a(a + 3d)+ …. +

a = first term l = last term d = common difference

There are n pairs of numbers that add up to (a + l)

Arithmetic Series = ½ n (a + l)

Arithmetic SeriesArithmetic Seriesa = first term l = last term d = common difference

Arithmetic Series = ½ n (a + l)

From last lesson, we know that the nth term (last term) is given by:

l = a + (n – 1) d

Arithmetic Series = ½ n (2a + (n – 1) d )

Arithmetic SeriesArithmetic Seriesa = first term l = last term d = common difference

Arithmetic Series = ½ n (a + l)

Arithmetic Series = ½ n (2a + (n – 1) d )

Why are both of these formulae useful?

Example 1Example 1Find the sum of the arithmetic series:11 + 15 + 19 + … + 107

l = a + (n – 1)d

n = 25Sum = ½ n (a + l)

a = 11 d = 4

From last lesson…

Solving…

l = 107

107 = 11 + 4(n – 1) Sub values in…

Sum = ½ 25 (11 + 107)

Sum = 1475

Using formula…

Sub values in…

ActivityActivity

Turn to page 42 of your textbook and answer

questions in Exercise E