Material Taken From: Mathematics for the international student Mathematical Studies SL Mal Coad, Glen Whiffen, John Owen, Robert Haese, Sandra Haese and Mark Bruce Haese and Haese Publications, 2004 AND Mathematical Studies Standard Level Peter Blythe, Jim Fensom, Jane Forrest and Paula Waldman de Tokman Oxford University Press, 2012

The dots means that the sequence Number Patterns Sequence: Terms: a list of numbers where there is a pattern the numbers in the sequence 3, 7, 11, 15, … The dots means that the sequence carries on infinitely

⭐An arithmetic sequence goes up, Describe the sequence: 14, 17, 20, 23, …. Write down the next two terms. - The numbers keep going up by 3 - 26, 29 ⭐An arithmetic sequence goes up, or down, in equal steps

Arithmetic Sequences The number added each time is called the common difference. The common difference is represented by the symbol d . The first term in a sequence is written as u1; the second term is u2; the third term is u1; and the nth term is un. The common difference (d) can always be obtained by taking any term and subtracting the previous term from it d = u2 – u1 or d = u12 – u11 Note that d can be either + or - .

In the arithmetic sequence: 3, 3. 5, 4, 4 In the arithmetic sequence: 3, 3.5, 4, 4.5, … What is the common difference? What is u1 and u3? d = 0.5 u1 = 3 u3 = 4 In the arithmetic sequence: 1, -2, -5, -8, … What is the common difference? What is u1 and u3? d = -3 u1 = 1 u3 = -5

Consider the pattern of marbles: Section 12B – Sequences of Numbers Consider the pattern of marbles: First layer has one blue marble Second layer has three pink marbles Third layer has five black marbles Fourth layer has seven green marbles

Consider the pattern of marbles: Let un represent the number of marbles in the nth layer: u1 = 1 u2 = 3 u3 = u4 = 5 7 The pattern could be continued forever…

The General Term un = u1 + (n – 1)d un represents the general term or nth term. The general term is defined for n = 1, 2, 3, 4, … For the marbles (and any arithmetic sequence) the general term is: un = u1 + (n – 1)d This is the formula used to determine the nth term in an arithmetic sequence

Here is a sequence of numbers. 2, 5, 8, 11, 14, 17, … Show that the sequence is an arithmetic sequence. Write down the common difference. Find the 10th term. Find the 25th term. A. The sequence goes up in equal sized steps of 3, thus it is arithmetic B. d = u3 – u2 = (8 – 5) = 3 C. un = u1 + (n – 1)d u10 = 2 + (10 – 1) x 3 = 29 D. un = u1 + (n – 1)d u25 = 2 + (25 – 1) x 3 = 74

Find the common difference, first term and general term un for an arithmetic sequence given that u3 = 8 and u8 = -17 Any arithmetic sequence can be written as: u1 u2 = u1 + 1d u3 = u2 + 1d = (u1 + 1d) + 1d = u1 + 2d u4 = u3 + 1d = (u1 + 2d) + 1d = u1 + 3d u3 = 8 = u1 + 2d AND u8 = -17 = u1 + 7d u1 = 8 – 2d AND u1 = - 17 – 7d 8 – 2d = - 17 – 7d d = – 5 un = u1 + (n – 1)d u1 = 8 – 2(-5) = 18 un = 18 + (n – 1)(-5) un = 23 – 5n

The second term of an arithmetic sequence is 1 and the seventh is 26 Find the first term and the common difference. Find the 100th term. u2 = u1 + 1d = 1 u7 = u1 + 6d = 26 u1 = 1 – 1d AND u1 = 26 – 6d 1 – d = 26 – 6d d = 5 u1 = 1 – 1(5) = - 4 un = u1 + (n – 1)d u100 = - 4 + (100 – 1)(5) u100 = 491

Insert four numbers between 3 and 12 so that all six numbers are in arithmetic sequence. If we are adding 4 numbers then 12 must be u6 Thus 3 + 5d = 12 d = 1.8 3, 4.8, 6.6, 8.4, 10.2, 12

Here is a sequence of numbers 6, 10, 14, …, 50 a Here is a sequence of numbers 6, 10, 14, …, 50 a. Write down the common difference. b. Find the number of terms in the sequence d = u2 - u1 = 10 – 6 = 4 un = 50 = u1 + (n - 1)d 50 = 6 + (n - 1)4 44 = (n - 1)4 11= n - 1 n = 12

The sum of the n terms of an arithmetic sequence is termed an arithmetic series and is written at Sn … un

The sum of the n terms of an arithmetic sequence is given by the two formulas: Use this form when you have the first and last term Use this form when you have the first term and the common difference OR

Find the sum of 4 + 7 + 10 + 13 + … to 50 terms. d = u2 – u1 = 7 – 4 = 3 or the common difference n = 50 or the number of terms u1 = 4 or the first term Sn = 𝟓𝟎 𝟐 (2(4) + (50 – 1)3) Sn = 25(8 + 147) = 25(155) = 3875

Find the sum of -6 + 1 + 8 + 15 + … + 141 d = u2 – u1 = 1 – (-6) = 7 or the common difference u1 = -6 or the first term un = 141 = u1 + (n – 1)d un = 141 or the last term 141 = -6 + (n – 1)7 147 = 7n + 7 154 = 7n n = 22 Sn = 𝟐𝟐 𝟐 (-6 + 141) Sn = 11(135) = 1485

The first five terms of an arithmetic sequence are shown below The first five terms of an arithmetic sequence are shown below. 2, 6, 10, 14, 18 … (a) Write down the sixth number in the sequence. (b) Calculate the 200th term. (c) Calculate the sum of the first 90 terms of the sequence. (a) 22 (b) 798 (c) 16200