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### Arithmetic

Sequences & Series

By: Jeffrey Bivin

Lake Zurich High School

Last Updated: April 28, 2006

Arithmetic Sequences

5, 8, 11, 14, 17, 20, … 3n+2, …

-4, 1, 6, 11, 16, … 5n – 9, . . .

11, 7, 3, -1, -5, … -4n + 15, . . .

Jeff Bivin -- LZHS

Find the nth term of an arithmetic sequence

First term is 8

Common difference is 3

an = a1 + d(n – 1)

an = 8 + 3(n – 1)

an = 8 + 3n – 3

an = 3n + 5

Jeff Bivin -- LZHS

Finding the nth term

First term is -6

common difference is 7

an = a1 + d(n – 1)

an = -6 + 7(n – 1)

an = -6 + 7n – 7

an = 7n - 13

Jeff Bivin -- LZHS

Finding the nth term

First term is 23

common difference is -4

an = a1 + d(n – 1)

an = 23 + -4(n – 1)

an = 23 -4n +4

an = -4n + 27

Jeff Bivin -- LZHS

Finding the 100th term

a1 = 5

d = 6

n = 100

5, 11, 17, 23, 29, . . .

an = a1 + d(n – 1)

a100 = 5 + 6(100 – 1)

a100 = 5 + 6(99)

a100 = 5 + 594

a100 = 599

Jeff Bivin -- LZHS

Finding the 956th term

a1 = 156

d = -16

n = 956

156, 140, 124, 108, . . .

an = a1 + d(n – 1)

a956 = 156 + -16(956 – 1)

a956 = 156 - 16(955)

a956 = 156 - 15280

a956 = -15124

Jeff Bivin -- LZHS

Find the Sum of the integers from 1 to 100

S100 = 1 + 2 + 3 +…+ 49 + 50 + 51 + 52 +…+ 98 + 99 + 100

S100 = 100 + 99 + 98 +…+ 52+51 + 50 + 49 +…+ 3 + 2 + 1

2S100 = 101+101+101+…+101+101+101+101+…+101+101+101

2S100 = 100 (101)

Jeff Bivin -- LZHS

Summing it up

Sn = a1 + (a1 + d) + (a1 + 2d) + …+ an

Sn = an + (an - d) + (an - 2d) + …+ a1

Jeff Bivin -- LZHS

Find the sum of the multiples of 3 between 9 and 1344

Sn = 9 + 12 + 15 + . . . + 1344

a1 = 9

an = 1344

d = 3

Jeff Bivin -- LZHS

Find the sum of the multiples of 7 between 25 and 989

Sn = 28 + 35 + 42 + . . . + 987

a1 = 28

an = 987

d = 7

Jeff Bivin -- LZHS

Find the sum of the multiples of 11 that are 4 digits in length

Sn = 10 01+ 1012 + 1023 + ... + 9999

a1 = 1001

an = 9999

d = 11

Jeff Bivin -- LZHS

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