Arithmetic Sequences & Partial Sums

MATH 109 - PrecalculusS. Rook

Overview

• Section 9.2 in the textbook:– Arithmetic sequences– Partial sums of arithmetic sequences

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Arithmetic Sequences

Arithmetic Sequences

• Arithmetic sequence: a sequence where the difference between ANY two successive terms is equal to the same constant value– i.e. ai+1 – ai = 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

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43,,8,5,2,1 n

2

3,,

2

7,3,

2

5,2

n

Arithmetic Sequences (Continued)

• The formula for the nth term of an arithmetic sequence is where a1 is the first term of the sequence and d is the difference between any two successive terms

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Arithmetic Sequences (Example)

Ex 1: Find the indicated term of the arithmetic sequence:a) Find the 12th term of the arithmetic sequence where the first term is -1 and the second term is 3

b) Find the 20th term of the arithmetic sequence where the first term is 2 and the fifth term is 4

c) Find the 50th term of the arithmetic sequence where the seventh term is -27 and the eighth term is -30

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Partial Sums of Arithmetic Sequences

Partial Sums of Arithmetic Sequences

• The nth partial sum of an arithmetic sequence is given by where a1 is the first term and an is the nth 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

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nn aan

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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

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25

1

67i

i

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 an that represents the price of the nth textbook purchased in the same orderb) Excluding taxes, find how much a customer will pay for purchasing 9 textbooks on the same order

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Summary

• After studying these slides, you should be able to:– Derive the formula for the nth 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)

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