Sequences

MATH 102Contemporary Math

S. Rook

Overview

• Section 6.6 in the textbook:– Arithmetic sequences– Geometric sequences

Arithmetic Sequences

Sequences

• Sequence: a list of numbers that follows some pattern. Each number in the list is referred to as a term. – Can be written as a1, a2, a3, …

– The nth term defines the pattern of the sequence• e.g. 1, 3, 5, 7, …, 2n – 1

• We will be examining two types of sequences:– Arithmetic– Geometric

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

5

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

6

dnaan 11

Sums of First n Terms of an Arithmetic Sequence

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

• Do not worry about deriving the formula – just know how to use it– e.g. What is the sum of the first 90 numbers (i.e.

1 – 90)?

7

nn aan

S 12

Arithmetic Sequences (Example)

Ex 1: For each arithmetic sequence, i) find an and ii) find the sum from terms 1 to an

a) 2, 8, 14, 20; a15

b) -6, -2, 2, 6; a22

Arithmetic Sequences (Example)

Ex 2: There is a pyramid of cans against the wall of a supermarket. There are 30 cans on the first row, 29 on the second row, 28 on the third row, and so on up to the thirtieth row where there is 1 can. How many total cans are in the stack?

Geometric Sequences

Geometric Sequences

• Geometric sequence: a sequence where the ratio of ANY two successive terms is equal to the same constant value

for all natural numbers i where r is known as the common ratio

– e.g.: a1 = 1 and r = 2

– e.g.: a1 = 4 and r = ½

11

i

i

a

ar 1

12,,8,4,2,1 n1

2

14,,

2

1,1,2,4

n

Geometric Sequences (Continued)

• The formula for the nth term of a geometric sequence is where a1 is the first term of the sequence and r is the common ratio

12

11

nn raa

Partial Sums of Finite Geometric Sequences

• The nth partial sum of a geometric sequence is given by where a1 is the first

term and r is the common ratio– Do not need to worry about deriving the formula– Just know how to use it• e.g. Find the sum of the first 15 terms of the geometric

series whose first term is 10 and second term is 5

13

1

11

r

raS

n

n

Geometric Sequences (Example)

Ex 3: For each geometric sequence, i) find an and ii) find the sum from terms 1 to an

a) 1, 3, 9, 27; a11

b) 3, 6, 12, 24; a9

Geometric Sequences (Example)

Ex 4: A ball is dropped from a height of 8 feet. The ball always bounces 7/8 of the distance from which it was dropped. What will be the height of the ball after the fifth bounce?

Arithmetic vs Geometric Sequences (Example)

Ex 5: Determine whether the sequence is arithmetic, geometric, or neither:

a) 1, 0, 1, 0, 1, …

b) 700, 750, 800, 850, …

c) -8, 2, -½ , …

Summary

• After studying these slides, you should know how to do the following:– Find the nth term of an arithmetic or geometric sequence– Find the sum of the first n terms of an arithmetic or

geometric sequence– Differentiate between an arithmetic or geometric series

• Additional Practice:– See problems in Section 6.6

• Next Lesson:– Linear Equations (Section 7.1)