Lake Zurich High School Arithmetic Sequences & Series By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Last Updated: April 28, 2006

Arithmetic Sequences 5, 8, 11, 14, 17, 20, … 3n+2, …

nth term of arithmetic sequence an = a1 + d(n – 1)

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

an = a1 + d(n – 1) an = -6 + 7(n – 1) an = -6 + 7n – 7 an = 7n - 13 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

an = a1 + d(n – 1) an = 23 + -4(n – 1) an = 23 - 4n + 4 an = -4n + 27 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

an = a1 + d(n – 1) a100 = 5 + 6(100 – 1) a100 = 5 + 6(99) Finding the 100th term 5, 11, 17, 23, 29, . . . an = a1 + d(n – 1) a100 = 5 + 6(100 – 1) a100 = 5 + 6(99) a100 = 5 + 594 a100 = 599 a1 = 5 d = 6 n = 100

an = a1 + d(n – 1) a956 = 156 + -16(956 – 1) a956 = 156 - 16(955) 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

Find the Sum of the integers from 1 to 100

Summing it up Sn = a1 + (a1 + d) + (a1 + 2d) + …+ an Sn = an + (an - d) + (an - 2d) + …+ a1

1 + 4 + 7 + 10 + 13 + 16 + 19 a1 = 1 an = 19 n = 7

4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 a1 = 4 an = 24 n = 11

Find the sum of the integers from 1 to 100 a1 = 1 an = 100 n = 100

Find the sum of the multiples of 3 between 9 and 1344 Sn = 9 + 12 + 15 + . . . + 1344 Jeff Bivin -- LZHS Jeff Bivin -- LZHS

Find the sum of the multiples of 7 between 25 and 989 Sn = 28 + 35 + 42 + . . . + 987 Jeff Bivin -- LZHS

Evaluate a1 = 16 an = 82 d = 3 n = 23 Sn = 16 + 19 + 22 + . . . + 82

Evaluate Sn = -29 - 31 - 33 + . . . - 199 a1 = -29 an = -199 d = -2

Find the sum of the multiples of 11 that are 4 digits in length an = 9999 d = 11 Sn = 10 01+ 1012 + 1023 + ... + 9999 Jeff Bivin -- LZHS

Review -- Arithmetic Sum of n terms nth term