Arithmetic SequencesArithmetic Sequences

Lesson 1.3Lesson 1.3

Arithmetic SequenceArithmetic Sequence

• This is a sequence in which the difference between each term and the preceding term is always constant.

• {10, 7, 4, 1, -2, -5,…}• Is {2, 4, 7, 11, 16,…} an arithmetic

sequence?• Recursive Form of arithmetic sequence

un = un-1 + dFor some constant d and all n ≥ 2

ExampleExample

If {un} is an arithmetic sequence with u1 = 2.5 and u2 = 6 as its first two terms

a. Find the common difference

b. Write the sequence as a recursive

function

c. Give the first six terms of the sequence

d. Graph the sequence

Explicit Form of Arithmetic SequenceExplicit Form of Arithmetic Sequence

• In an arithmetic sequence {un} with common difference d, un = u1 + (n-1)d for every n ≥ 1.

• If u1 = -5 and d = 3 we can find the explicit form by, un = -5 + (n-1)3 = -5 + 3n – 3… leaving us with the explicit form of 3n - 8

ExampleExample• If we wanted to know the 38th term of the

arithmetic sequence whose first three terms are 15, 10, and 5, how would we do that?

Here’s HowHere’s How un = u1 + (n-1)d

= 15 + (38-1)(-5)

= 15 + (-5)(37)

= 15 + -185

= -170

• Lets look at example 6 on page 24 because it is far too exhaustive to write down!

Summation NotationSummation Notation

1 2 31

...m

k mk

c means c c c c

5

1

( 7 3 )n

n

What is the sum of this sequence?

Graphing Calculator ExplorationGraphing Calculator Exploration

• We are going to use the sum sequence key on our graphing calculators

• Find the sum of this little diddy

100

1

5 6n

n

Partial Sums of Arithmetic SequencesPartial Sums of Arithmetic Sequences

• If {un} is an arithmetic sequence with common difference d, then for each positive integer k, the kth partial sum can be found by using either of the following formulas

11

11

1.2

( 1)2.

2

k

n kn

k

nn

ku u u

k ku ku d

There is a proof on this on page 27…if anybody really cares

ExampleExample

• Find the 14th partial sum of the arithmetic sequence 21, 15, 9, 3,…

U14 = u1 + (14 – 1)(-6)

= 21 + (13)(-6)

= 21 + (-78)

= -57

12

1

12(21 57)2

7( 36)

252

nn

u

Find the Sum of all multiples of 4 Find the Sum of all multiples of 4 from 4 to 404!from 4 to 404!

• We know that we are adding 4 + 8 + …, so 4x1, 4x2, 4x3, … and we can take 404 ÷ 4 to get the 101 term.

• What this means is there is 101 multiples of 4 in between 4 and 404

• u1=4, k=101, and u101 = 404! Use form 1

101

1

101 101(4 404) (408) 20,604

2 2nn

u

Here is a little story about LarryHere is a little story about Larry• Larry owns an automobile dealership. He spends $18,000

on advertising during the first year, and he plans to increase his advertising expenditures by $1400 in each subsequent year. How much will Larry spend on advertising during the first 9 years?

9

9

9

1

$18,000 (9 1)(1400)

18,000 1400(8)

18,000 11,200

$29,200

:

9 9($18,000 $29,200) ($47,200)2 2

$212,400

nn

u

u

Next

u

Now…Get To Work Slackers!Now…Get To Work Slackers!