12.2 Arithmetic Sequences

©2001 by R. Villar

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Arithmetic Sequences

Arithmetic Sequence: sequence whose consecutive terms have a common difference.

Example: 3, 5, 7, 9, 11, 13, ...

The terms have a common difference of 2.

The common difference is the number d.

Example: Is the sequence arithmetic? –45, –30, –15, 0, 15, 30 Yes, the common difference is 15

How do you find any term in this sequence?To find any term in an arithmetic sequence, use the

formula an = a1 + (n – 1)dwhere d is the common difference.

Example: Find a formula for the nth term of the arithmetic sequence in which the common difference is 5 and the first

term is 3.

an = a1 + (n – 1)d a1 = 3 d = 5an = 3 + (n – 1)5

Example: If the common difference is 4 and the fifth term is 15, what is the 10th term of an arithmetic sequence?

an = a1 + (n – 1)d We need to determine what the first term is...

d = 4 and a5 = 15a5 = a1 + (5 – 1)4 = 15a1 = –1

a10 = –1 + (10 – 1)4 a10 = 35

Determine whether each sequence appears to be an arithmetic sequence. If

so, find the common difference and the next three terms in the sequence.

A. 12, 8, 4, 0, …

B. 1, 4, 9, 16, …

Find the indicated term of each arithmetic sequence.

A. 22nd term: 5, 2, -1, -4, …

B. 15th term: a1 = 7; d = 3

a. Find the next three terms of thearithmetic sequence 0, 9, 18, 27,…

b. Find the 7th term of the arithmeticsequence 10, 23, 36,...

Which of the following is the 24th term of the

arithmetic sequence 3, 2, 7, 12,...?

A 62 B 92 C 112 D 162

100 Category 1

Answer

3, 6, 9, ___, ___,___

Answer 100 Category 1

3, 6, 9, 12, 15, 18

200 Category 1

Answer

20, 12, 4 , ___, ___,___

Answer 200 Category 1

20, 12, 4 , -4, -12, -20

300 Category 1

Answer

800, 400, 200, , 800, 400, 200, , ___, ___,______, ___,___

Answer 300 Category 1

800, 400, 200, 100, 800, 400, 200, 100, 50,2550,25

400 Category 1

Answer

3, 5, 8, 12, ___, 3, 5, 8, 12, ___, ___,______,___

Answer 400 Category 1

3, 5, 8, 12, 17, 3, 5, 8, 12, 17, 23,3023,30

500 Category 1

Answer

1,1,2,3,5,8 , ___, 1,1,2,3,5,8 , ___, ___,______,___

Answer 500 Category 1

1,1,2,3,5,8 , 13, 1,1,2,3,5,8 , 13, 21,3421,34

100 Category 2

Answer

Write the rule if Write the rule if aa11= 10 d = 3= 10 d = 3

Answer 100 Category 2

Write the rule if Write the rule if aa11= 10 d = 3= 10 d = 3aann = 7 + 3n = 7 + 3n

200 Category 2

Answer

Write the rule ifWrite the rule ifaa33= 15 d = -2= 15 d = -2

Answer 200 Category 2

Write the rule ifWrite the rule ifaa33= 15 d = -2= 15 d = -2aann = 21 - 2n = 21 - 2n

300 Category 2

Answer

Write the first 5 Write the first 5 terms for aterms for ann=6+5n=6+5n

Answer 300 Category 2

Write the first 5 Write the first 5 terms for aterms for ann=6+5n=6+5n11, 16, 21, 26, 3111, 16, 21, 26, 31

400 Category 2

Answer

Find aFind a2020 for for 10, 6, 2, -2... 10, 6, 2, -2...

Answer 400 Category 2Find aFind a2020 for for

10, 6, 2, -2… 10, 6, 2, -2…aann =14 - 4n =14 - 4n

so aso a20 = 14 - 4(20) = -6620 = 14 - 4(20) = -66

500 Category 2

Answer

Find the rule if aFind the rule if a33 = = 5 and a5 and a77 = 9 = 9

Answer 500 Category 2

Find the rule if aFind the rule if a33 = 5 = 5 and aand a77 = 9 = 9

__,__,5,__,__,__,9__,__,5,__,__,__,9d = (9-5)/(7-3) =1d = (9-5)/(7-3) =1

aann = 2 +1n = 2 +1n