sequences and series (4) learn what is meant by an arithmetic series
TRANSCRIPT
![Page 1: Sequences and Series (4) Learn what is meant by an ARITHMETIC SERIES](https://reader036.vdocuments.site/reader036/viewer/2022082710/56649e245503460f94b11d30/html5/thumbnails/1.jpg)
Sequences and Series (4)Sequences and Series (4)
Learn what is meant by an Learn what is meant by an
ARITHMETIC SERIESARITHMETIC SERIES
![Page 2: Sequences and Series (4) Learn what is meant by an ARITHMETIC SERIES](https://reader036.vdocuments.site/reader036/viewer/2022082710/56649e245503460f94b11d30/html5/thumbnails/2.jpg)
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?
![Page 3: Sequences and Series (4) Learn what is meant by an ARITHMETIC SERIES](https://reader036.vdocuments.site/reader036/viewer/2022082710/56649e245503460f94b11d30/html5/thumbnails/3.jpg)
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)
![Page 4: Sequences and Series (4) Learn what is meant by an ARITHMETIC SERIES](https://reader036.vdocuments.site/reader036/viewer/2022082710/56649e245503460f94b11d30/html5/thumbnails/4.jpg)
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
![Page 5: Sequences and Series (4) Learn what is meant by an ARITHMETIC SERIES](https://reader036.vdocuments.site/reader036/viewer/2022082710/56649e245503460f94b11d30/html5/thumbnails/5.jpg)
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.
![Page 6: Sequences and Series (4) Learn what is meant by an ARITHMETIC SERIES](https://reader036.vdocuments.site/reader036/viewer/2022082710/56649e245503460f94b11d30/html5/thumbnails/6.jpg)
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)
![Page 7: Sequences and Series (4) Learn what is meant by an ARITHMETIC SERIES](https://reader036.vdocuments.site/reader036/viewer/2022082710/56649e245503460f94b11d30/html5/thumbnails/7.jpg)
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 )
![Page 8: Sequences and Series (4) Learn what is meant by an ARITHMETIC SERIES](https://reader036.vdocuments.site/reader036/viewer/2022082710/56649e245503460f94b11d30/html5/thumbnails/8.jpg)
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?
![Page 9: Sequences and Series (4) Learn what is meant by an ARITHMETIC SERIES](https://reader036.vdocuments.site/reader036/viewer/2022082710/56649e245503460f94b11d30/html5/thumbnails/9.jpg)
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…
![Page 10: Sequences and Series (4) Learn what is meant by an ARITHMETIC SERIES](https://reader036.vdocuments.site/reader036/viewer/2022082710/56649e245503460f94b11d30/html5/thumbnails/10.jpg)
ActivityActivity
Turn to page 42 of your textbook and answer
questions in Exercise E