SECTION 7.2SECTION 7.2

ARITHMETIC SEQUENCESARITHMETIC SEQUENCES

ARITHMETIC SEQUENCESARITHMETIC SEQUENCES

(a) (a) 5, 9, 5, 9, 13, 17, 13, 17,

21,21,2525

(b) (b) 2, 2.5, 3, 2, 2.5, 3, 3.5, 3.5,

4,4, 4.54.5

(c) (c) 8, 5, 2, 8, 5, 2, - 1, - 1,

- 4,- 4, - 7- 7

Adding 4Adding 4

Adding .5Adding .5

Adding - 3Adding - 3

Arithmetic Sequences have a Arithmetic Sequences have a “common difference”.“common difference”.

(a) 4(a) 4 (b) .5(b) .5 (c) (c) - 3- 3

ARITHMETIC SEQUENCE

RECURSION FORMULA

ARITHMETIC SEQUENCE

RECURSION FORMULAaa n n = a = a n - 1n - 1 + d + d

This formula relates each term This formula relates each term in the sequence to the in the sequence to the previous term in the sequence.previous term in the sequence.

aa n n = a = a n - 1n - 1 + 4 + 4

bb n n = b = b n - 1n - 1 + .5 + .5 cc n n = c = c n - n -

11 - 3 - 3

EXAMPLE:EXAMPLE:

Given that eGiven that e 1 1 = 4 and the = 4 and the recursion formula erecursion formula e n n = e = e n - 1n - 1 + + 0.3, determine the first five 0.3, determine the first five terms of the sequence { eterms of the sequence { e n n }.}.ee 1 1 = 4 = 4

ee 2 2 = 4 + .3 = 4.3 = 4 + .3 = 4.3ee 3 3 = 4.3 + .3 = = 4.3 + .3 = 4.64.6ee 4 4 = 4.6 + .3 = = 4.6 + .3 = 4.94.9ee 5 5 = 4.9 + .3 = = 4.9 + .3 = 5.25.2

Recursion Formulas have a big Recursion Formulas have a big disadvantage.disadvantage.

In the last example, what In the last example, what would happen if we needed to would happen if we needed to know the value of the 291st know the value of the 291st term?term?

Explicit Formulas are much Explicit Formulas are much better for finding nth terms.better for finding nth terms.

ARITHMETIC SEQUENCE EXPLICIT

FORMULA

ARITHMETIC SEQUENCE EXPLICIT

FORMULA

aa 1 1

aa11

aa 2 2

aa11

dd

aa 3 3

aa11

dd

dd

aa 4 4

aa11

dd dd

dd

aa n n

aa11

dd

1 d1 d

2 d’s2 d’s3 d’s3 d’s n-1 n-1

d’sd’s

a a nn = a = a 1 1 + (n - 1) d + (n - 1) d

Example:Example: Determine e Determine e 291291 for for the arithmetic sequence with the arithmetic sequence with ee11 = 4 and common = 4 and common difference d = 0.3difference d = 0.3

e e 291 291 = 4 + (291 - 1) (0.3) = 4 + (291 - 1) (0.3)

9191

ARITHMETIC SEQUENCE EXPLICIT

FORMULA

ARITHMETIC SEQUENCE EXPLICIT

FORMULA

SUMS OF ARITHMETIC SEQUENCES

SUMS OF ARITHMETIC SEQUENCES

1 + 2 + . . . + 49 + 50 + 51 + 52 + . . . 1 + 2 + . . . + 49 + 50 + 51 + 52 + . . . + 99 + 100+ 99 + 100

50 PAIRS OF 10150 PAIRS OF 101

50(101) = 505050(101) = 5050

ARITHMETIC SEQUENCE SUM

FORMULA

ARITHMETIC SEQUENCE SUM

FORMULA

S = na + a

2n1 n

The sum of n terms of an The sum of n terms of an arithmetic sequence is n arithmetic sequence is n times the average of the first times the average of the first and last terms to be added.and last terms to be added.

EXAMPLE:EXAMPLE:

Determine the sum of the Determine the sum of the first 200 terms of the first 200 terms of the arithmetic sequence { aarithmetic sequence { a n n } } with awith a 1 1 = - 5 and d = 3. = - 5 and d = 3.

First, we must find a First, we must find a 200200

a a 200200 = - 5 + (199)(3) = 592 = - 5 + (199)(3) = 592

EXAMPLE:EXAMPLE:

aa 1 1 = - 5 = - 5 and and a a 200200 = 592 = 592

S = 200 - 5 + 592

2200

58, 70058, 700

FINDING THE NUMBER OF TERMS IN A

SEQUENCE

FINDING THE NUMBER OF TERMS IN A

SEQUENCE

4, 9, 14, 19, . . . , 644, 9, 14, 19, . . . , 64

Just add 5 on the calculator Just add 5 on the calculator until you get to 64 and see until you get to 64 and see how many terms there are how many terms there are

in the sequence.in the sequence.

OR…OR…

FINDING THE NUMBER OF A TERMS IN A

SEQUENCE

FINDING THE NUMBER OF A TERMS IN A

SEQUENCE

a - a

d + 1n 1

1 + 5

4 - 64

EXAMPLEEXAMPLE

(2k - 1) = k = 1

50

(2 - 1) + (4 - 1) + (6 - 1) + . . . + (2 - 1) + (4 - 1) + (6 - 1) + . . . + (100 - 1)(100 - 1)

1 + 3 + 5 + . . . + 1 + 3 + 5 + . . . + 99 99

This is the sum of an arithmetic This is the sum of an arithmetic sequence sequence

EXAMPLEEXAMPLE

(2k - 1) = k = 1

50

1 + 3 + 5 + . . . 1 + 3 + 5 + . . . + 99+ 99

Recall: S = na + a

2n1 n

S = 501 + 99

2 = 250050

CONCLUSION OF SECTION 7.2CONCLUSION OF SECTION 7.2