1. Arithmetic Sequences & Partial Sums MATH 109 - Precalculus S. Rook

2. Overview Section 9.2 in the textbook: – Arithmetic sequences – Partial sums of arithmetic sequences 2

3. Arithmetic Sequences

4. Arithmetic sequence: a sequence where the difference between ANY two successive terms is equal to the same constant value – i.e. a i+1 – a i = d for every natural number i where d is the difference e.g. starts at -1 with a difference of 3 e.g. starts at 2 with a difference of ½ 4

5. Arithmetic Sequences (Continued) The formula for the n th term of an arithmetic sequence is where a 1 is the first term of the sequence and d is the difference between any two successive terms 5

6. Arithmetic Sequences (Example) Ex 1: Find the indicated term of the arithmetic sequence: a) Find the 12 th term of the arithmetic sequence where the first term is -1 and the second term is 3 b) Find the 20 th term of the arithmetic sequence where the first term is 2 and the fifth term is 4 c) Find the 50 th term of the arithmetic sequence where the seventh term is -27 and the eighth term is -30 6

7. Partial Sums of Arithmetic Sequences

8. The n th partial sum of an arithmetic sequence is given by where a 1 is the first term and a n is the n th term – Also known as an Arithmetic Series Do not worry about deriving the formula – just know how to use it VERY important to know the difference between sequences and series: – A sequence is a LIST of TERMS – A series is a SUM of the terms of a sequence 8

9. Partial Sums of Arithmetic Sequences (Example) Ex 2: Find the indicated partial sum of the arithmetic sequence: a) b) n = 16; first three terms are ½, -¼, -1 9

10. Partial Sums of Arithmetic Sequences – Application (Example) Ex 3: A bookseller has an offer where the first copy of a particular textbook is sold at full price which is $100. Subsequent copies of the same textbook bought in the same order will be discounted $3.00 with a limit of ten textbooks per order. (i.e. first costs $100, second costs $97, third costs $94, etc) a) Write an arithmetic sequence a n that represents the price of the n th textbook purchased in the same order b) Excluding taxes, find how much a customer will pay for purchasing 9 textbooks on the same order 10

11. Summary After studying these slides, you should be able to: – Derive the formula for the n th term of an arithmetic sequence given at least two terms – Calculate the nth partial sum of an arithmetic sequence – State the difference between sequences and series Additional Practice – See the list of suggested problems for 9.2 Next lesson – Geometric Sequences & Series (Section 9.3) 11