Notes 3.4

Arithmetic Sequences

I. Sequences and Terms• Sequence: a list of numbers

in a specific order.

1, 3, 4, 7, 10, 16

• Term: each number in a sequence

Sequence

Terms

A sequence is arithmetic if the differences between consecutive terms are the same.

4, 9, 14, 19, 24, . . .

9 – 4 = 5

14 – 9 = 5

19 – 14 = 5

24 – 19 = 5

arithmetic sequence

The common difference, d, is 5.

II. Arithmetic Sequences- Definition and Formulas

FYI: Common differences can be negative.

A. Definition

How do I know if it is an arithmetic sequence?

• Look for a common difference between consecutive terms

Ex: 2, 4, 8, 16 . . . Common Difference?

2+2 = 4

4+ 4 = 8

8 + 8 = 16

No. The sequence is NOT Arithmetic

Ex: 48, 45, 42, 39 . . . Common Difference?

48 - 3 = 45

45 - 3 = 42

42 - 3 = 39

Yes. The sequence IS Arithmetic. d = -3

Example

• Find the next three terms in the arithmetic sequence: 2, 5, 8, 11, 14, __, __, __

• 2, 5, 8, 11, 14, 17, 20, 23

• The common difference is?

• 3!!!

B. Important Formulas for Arithmetic Sequence:

• Recursive Formula • Explicit Formula

an = an – 1 + d an = a1 + (n - 1)d

Where:

an is the nth term in the

sequence

a1 is the first term

n is the number of the term

d is the common difference

III. Finding terms given the formula

• Given the following formulas, find the first four terms.

Ex 2:

t1 = 2

tn = tt-1 -6

2, -4, -10, -16

Ex 3:

t1 = -3

tn = tt-1 +4

-3, 1, 5, 9

Ex 1:

t1 = 8

tn = tt-1 +5

8,13, 18, 23

A. Given Recursive

• Ex 1: Given tn = 5n-2, find the first four terms.

3, 8, 13, 18

B. Given Explicit

For the first term,

substitute 1 for n. t1 = 5(1)-2

t1 = 3 Do the same to

find the second,

third, and fourth

term.

t2 = 5(2)-2

t2 = 8

t3 = 5(3)-2

t3 = 13

t4 = 5(4)-2

t4 = 18

• Given tn = -3n+1 find the first four terms.

-2, -5, -8, -11

B. Given Explicit

For the first term,

substitute 1 for n. t1 = -3(1)+1

t1 = -2 Do the same to

find the second,

third, and fourth

term.

t2 = -3(2)+1

t2 = -5

t3 = -3(3)+1

t3 = -8

t4 = -3(4)+1

t4 = -11

Ex 1: Write the explicit and recursive formula for each sequence

2, 4, 6, 8, 10

First term: a1 = 2

difference = 2

an = an – 1 + dan = an – 1 + 2, where

a1 = 2Recursive:

an = a1 + (n-1)d an = 2 + (n-1)2Explicit:

IV. Write the formulas given the sequence.

Ex 2: Write the explicit and recursive formula for each sequence

7, 11, 15, 19…

First term: a1 = 7

difference = 4

an = an – 1 + dan = an – 1 + 4, where

a1 = 7Recursive:

an = a1 + (n-1)d an = 7 + (n-1)4Explicit:

IV. Write the formulas given the sequence.

V. Recursive to Explicit• Given the recursive formula, write the explicit

formula for the sequence.

Ex 1:

a1 = 8

an = an-1 +5

an = a1 + (n-1)d

an = 8 + (n-1)5

Ex 2:

a1 = -5

an = aa-1 +2

an = a1 + (n-1)d

an = -5 + (n-1)2

VI. Explicit to Recursive

an = an – 1 + d

an = an – 1 + 4, where

a1 = 6

Ex 1: an = 4n + 2

a1 = 4(1) + 2

a1 = 6

a2 = 4(2) + 2

a2 = 10

d = 4

an = an – 1 + d

an = an – 1 + 3, where

a1 = -5

Ex 2: an = 3n - 8

a1 = 3(1) - 8

a1 = -5

a2 = 3(2) - 8

a2 = -2

d = 3

VII. Finding the nth term.

Ex 1: Find the 25th term in the sequence of 5, 11, 17, 23, 29 . . .

an = a1 + (n-1)d

a25 = 5 + (24)6 =149

difference = 6

a25 = 5 + (25 -1)6

Start with the explicit sequence formula

Find the common difference

between the values.

Plug in known values

Simplify

Explicit Arithmetic Sequence Problem

Ex 2: Find the 17th term of the arithmetic sequence: 26, 13, 0, -13

an = a1 + (n-1)d

a25 = 26 + (16)(-13) =-182

difference = -13

a25 = 26 + (17 -1)(-13)

Start with the explicit sequence formula

Find the common difference

between the values.

Plug in known values

Simplify

Ex. 3: Suppose you have saved $75 towards the purchase of a new iPad. You plan to save at least $12 from mowing your neighbor’s yard each week. In all, what is the minimum amount of money you will have in 26 weeks?

an = a1 + (n-1)d

a26 = 75 + (26)12 =$387

difference = 12

a26 = 75 + (27 -1)12

Start with the explicit sequence formula

Find the common difference

between the values. You will save

$12 a week so this is your difference.

Plug in known values

Simplify

WAIT: Why 27 and not 26 for n ?

The first term in the sequence, 75, came before the weeks started (think of it

as week 0). Therefore you want one more week in your formula to account

for the $75 that you had before you started saving.