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

An arithmetic sequence: is a sequence in which each term differs from the previous one by the same fixed number. » 4, 7, 10, 13, …. » 15, 9, 3, -3, …. Section 12C – Arithmetic Sequences Algebraic Definition : {u n } is arithmetic if (and only if) u n+1 – u n = d for all positive integers n and where d is a constant (the common difference).

u n = u 1 + d(n – 1) The General Term of an arithmetic sequence For an arithmetic sequence with the first term u 1 and common difference d the general term (u n ) is given by:

1) Consider the sequence 2, 9, 16, 23, 30…. a) Show that the sequence is arithmetic. b) Find the formula for the general term u n. c) Find the 100th term of the sequence. d) Is (i) 828 (ii) 2341 a member of the sequence?

2) Find k given that 3k + 1, k and -3 are consecutive terms of an arithmetic sequence.

3) The first term of an arithmetic sequence is - 16 and the eleventh term is 39. Calculate the value of the common difference.

4) Find the general term u n for an arithmetic sequence given that u 3 = 8 and u 8 = -17

5) Insert four numbers between 3 and 12 so that all six numbers are in arithmetic sequence.

CHALLENGE: Sum the numbers from 1 to 100

Once upon a time…

When Carl Friedrich Gauss was ten, he was admitted to a beginning class in arithmetic. The teacher gave the class a problem to sum the numbers from 1 to 100, which he expected would take the students most of the class time. In those days, it was custom for the students to stack their slates on the teacher’s table in the order in which they were completed. The problem had barely been stated when Gauss placed his slate upon the table. When the answers were checked, only Gauss’ showed the correct answer.

CHALLENGE: Sum the numbers from 1 to

… … What do each of these pairs sum to? 101 How many pairs do you have? 50

Carl Friedrich Gauss German mathematician & scientist “Prince of Mathematicians” or “the greatest mathematician since antiquity.” Gauss’ most important discovery was how to construct a regular 17-gon by straight-edge and compass. On 9 October, 1805 Gauss married Johanna Ostoff. In 1809 his wife died after giving birth to their second son, who was to die soon after her. In 1810 Gauss was married for a second time, to Minna the best friend of Johanna. Later in life his health deteriorated slowly, and Gauss died in his sleep early in the morning of 23 February, 1855.

Carl Friedrich Gauss

6) For the sequence 1, 4, 9, 16, 25, … a)Write down an expression for S n. b)Find S n for n = 1, 2, 3, 4 and 5. Section 12G - Series A Series: is the addition of terms of a sequence.  u 1 + u 2 + u 3 + … + u n The sum of the series is the result when all the terms of the series are added.

7) Find the sum of … to 50 terms. Remember: S n = n/2(u 1 + u n ) or S n = n/2(2u 1 + (n – 1)d)

8) Find the sum of … + 141

(a) The first term of an arithmetic sequence is –16 and the eleventh term is 39. Calculate the value of the common difference. (b) The third term of a geometric sequence is 12 and the fifth term is 163. All the terms in the sequence are positive. Calculate the value of the common ratio.

9) The first five terms of an arithmetic sequence are 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.

Homework Pg – Section 12C – #1, 3, 5ace, 6a, 7 Pg – Section 12G.2 – #1abc, 2bc, 3ab, 4, 7