Sequences and Series (4) Learn what is meant by an ARITHMETIC SERIES

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 Series An Arithmetic series is the sum of the terms in an Arithmetic sequence. Eg. 1, 2, 3, 4… (Arithmetic sequence) … (Arithmetic series)

Back to the pyramid… We wanted to work out the sum of: … … If we write it out in reverse we get… ….. How many times do we add 101 together?101 x 100 = What do we need to do to this answer?10100 / 2 = 5050

Activity 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 Series Work 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 terml = last termd = common difference There are n pairs of numbers that add up to (a + l) Arithmetic Series = ½ n (a + l)

Arithmetic Series a = first terml = last termd = common difference Arithmetic Series = ½ n (a + l) From last lesson, we know that the n th term (last term) is given by: l = a + (n – 1) d Arithmetic Series = ½ n (2a + (n – 1) d )

Arithmetic Series a = first terml = last termd = common difference Arithmetic Series = ½ n (a + l) Arithmetic Series = ½ n (2a + (n – 1) d ) Why are both of these formulae useful?

Example 1 Find the sum of the arithmetic series: … l = a + (n – 1)d n = 25 Sum = ½ n (a + l) a = 11d = 4 From last lesson… Solving… l = = (n – 1) Sub values in… Sum = ½ 25 ( ) Sum = 1475 Using formula… Sub values in…

Activity Turn to page 42 of your textbook and answer questions in Exercise E