Arithmetic Sequences & Series Last Updated: October 11, 2005

Arithmetic Progression 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 n th term

n th term of arithmetic sequence T n = a + d(n – 1) Jeff Bivin -- LZHS a = First term d = common difference n = number of terms. Common difference = the difference between two consecutive terms in a sequence. d = T n – T n-1

8, 11, 14, 17, 20, … First term is 8 Common difference is 3 T n = a + d(n – 1) T n = 8 + 3(n – 1) T n = 8 + 3n – 3 T n = 3n + 5 Jeff Bivin -- LZHS Find the n th term of the following AP.

Finding the n th term  6, 1, 8, 15, 22, … First term is -6 common difference is 7 T n = a + d(n – 1) T n = (n – 1) T n = n – 7 T n = 7n - 13 Jeff Bivin -- LZHS

Finding the n th term 23, 19, 15, 11, 7, … First term is 23 common difference is -4 T n = a + d(n – 1) T n = (n – 1) T n = 23 -4n + 4 T n = -4n + 27 Jeff Bivin -- LZHS

Finding the 956 th term 156, 140, 124, 108,... T n = a + d(n – 1) T 956 = (956 – 1) T 956 = (955) T 956 = T 956 = a 1 = 156 d = -16 n = 956 Jeff Bivin -- LZHS

Finding the 100 th term 5, 11, 17, 23, 29,... T n = a + d(n – 1) T 100 = 5 + 6(100 – 1) T 100 = 5 + 6(99) T 100 = T 100 = 599 a = 5 d = 6 n = 100 Jeff Bivin -- LZHS

Finding the number of terms in the AP 10, 8, 6, 4, 2, T n = a + d(n – 1) -24 = 10 -2(n – 1) -34 = -2(n – 1) 17 = n-1 n = 18 a = 10 d = -2 Tn = -24 Jeff Bivin -- LZHS

The 5 th term of an AP is 13 and the 13 th term is -19. Find the first term & the common difference. T 5 = a + 4d = 13……..(1) T 13 = a + 12d = -19……….(2) (2) – (1): 8d = d = - 32 d = -4 Substitute d = -4 into (1): a + 4(-4) = 13 a – 16 = 13 a = 29 Jeff Bivin -- LZHS

Problem solving In a race, a swimmer takes 35 seconds to swim the first 100 m, 39 seconds to swim the second 100m and 43 seconds to swim the third 100 m. If he continues to swim in this manner, how long does he take to finish the 10 th lap of 100m.

Summing it up S n = a 1 + (a 1 + d) + (a 1 + 2d) + …+ a n S n = a n + (a n - d) + (a n - 2d) + …+ a 1 Jeff Bivin -- LZHS

a 1 = 1 a n = 19 n = 7 Jeff Bivin -- LZHS

a 1 = 4 a n = 24 n = 11 Jeff Bivin -- LZHS

Find the sum of the integers from 1 to 100 a 1 = 1 a n = 100 n = 100 Jeff Bivin -- LZHS

Find the sum of the multiples of 3 between 9 and 1344 a 1 = 9 a n = 1344 d = 3 S n = Jeff Bivin -- LZHS

Find the sum of the multiples of 7 between 25 and 989 a 1 = 28 a n = 987 d = 7 S n = Jeff Bivin -- LZHS

Evaluate a 1 = 16 a n = 82 d = 3 n = 23 S n = Jeff Bivin -- LZHS

Evaluate a 1 = -29 a n = -199 d = -2 n = 86 S n = Jeff Bivin -- LZHS

Find the sum of the multiples of 11 that are 4 digits in length a 1 = 1001 a n = 9999 d = 11 S n = Jeff Bivin -- LZHS

Review -- Arithmetic n th term Sum of n terms Jeff Bivin -- LZHS