10.2Arithmetic Sequences

Date: ____________

Arithmetic Sequence

• Sequence in which each term after the first is obtained by adding a fixed number, called the difference, to the previous term.

5, 8, 11, 14, 17, . . .

+3 +3 +3 +3

16, 14, 12, 10, 8, . . .

-2 -2 -2 -2

Common difference is 3.

(d = 3)

Common difference is -2.

(d = -2)

Decide if each sequence is an arithmetic sequence. If yes, find the

common difference.

-5, -1, 3, 7, 11,... Yes. d = 4

4, 5, 7, 10, 14,… No.

1, 4, 8, 12, 16,… No.

-4, -7, -10, -13, -16,… Yes. d = -3

Arithmetic Sequence

an = a1 + d(n − 1)

an = nth term of the sequence

a1 = first term

n = # of terms

d = common difference

Find an and a20.

a1 = 7 d = 5

an = 7 + 5(n − 1)

an = 7 + 5n – 5

an = 2 + 5n

a20 = 2 + 5(20)

a20 = 102

an = a1 + d(n − 1)

Find an and a25.

48, 53, 58, 63,… an = a1 + d(n − 1)

48 5an = 48 + 5(n – 1)

an = 48 + 5n – 5

an = 43 + 5n

a25= 43 + 5(25)

a25 = 168

Find an and a25.

-21, -39, -57, -75,… an = a1 + d(n − 1)

-21 -18

an = -21 – 18(n – 1)

an = -21 – 18n + 18

an = -3 – 18n

a25= -3 – 18(25)

a25 = -453

Find an and a20.

a17 = 22 d = -4

an = a1 + d(n − 1)

22 = a1 – 4(17 − 1)

22 = a1 – 4(16)

22 = a1 – 64

86 = a1

an = 86 – 4(n − 1)

an = 86 – 4n +4

an = 90 – 4n

a20 = 90 – 4(20)

a20 = 10

Find an and a13.

a15 = 10 a20 = 25 d =25 – 10

20 – 15 =

15

5 = 3

an = a1 + d(n − 1)

10 = a1 + 3(15 − 1)

10 = a1 + 3(14)

10 = a1 + 42

-32 = a1

an = -32 + 3(n − 1)

an = -32 + 3n – 3

an = -35 + 3n

a13 = -35 + 3(13)

a13 = 4

Find an and a13.

a12 = -23 a27 = 37 d =37 − ‾23

27 – 12

=60

15 = 4

an = a1 + d(n − 1)

-23 = a1 + 4(12 − 1)

-23 = a1 + 4(11)

-23 = a1 + 44

-67 = a1

an = -67 + 4(n − 1)

an = -67 + 4n – 4

an = -71 + 4n

a13 = -71 + 4(13)

a13 = -19

Sum of a Finite Arithmetic Sequence

2

+= 1 n

n

aanS ( )

Find the sum of the first 10 terms of the sequence if a1 = -16 and a10 = 20

2

20+1610=10S ( ) S10 = 20

Find the sum of the first 42 terms of the sequence if a1 = 7 and a42 = 239

( )2

239+742=42S

S42 = 5166

2

+= 1 n

n

aanS ( )

Find the sum of the first 100 terms of the sequence if a1 = 5 and d = 3.

2

+100= 1001

100

aaS ( ) an = a1 + d(n − 1)

a100 = 5 + 3(100 − 1)

a100 = 3022

302+5100=100S ( )

S100 = 15,350

Find the sum of the first 24 terms of the sequence if a1 = -4 and d = -6.

2

+24= 241

24

aaS ( ) an = a1 + d(n − 1)

a24 = -4 – 6(24 − 1)

a24 = -1422

421+424=24S ( )S24 = -1752

Find the sum of the first 50 terms of the sequence 34, 45, 56, 67, 78,…

2

+50= 501

50

aaS ( ) a50 = 34 + 11(50 − 1)

a50 = 573

2

573+3450=50S ( )

S50 = 15,175

an = a1 + d(n − 1)

Find the sum of the first 20 terms of the sequence 12, 18, 24, 30, 36,…

2

+20= 201

20

aaS ( ) a20 = 12 + 6(20 − 1)

a20 = 126

2

126+1220=20S ( )

S20 = 1380

an = a1 + d(n − 1)