Write the next three numbers in each pattern

Write the next three numbers in each pattern. 1) 15, 11, 7, 3, -1, … 2) -12, -7, -2, 3, 8, … 3) 10, 11, 13, 16, 20, … 4) 2, 6, 18, 54, 162, … 5) 96, 48, 24, 12, 6, … 6) 1, 4, 9, 16, 25, … Sequences and Series

Sequences A sequence is an ordered list of numbers, called terms. An infinite sequence continues without end while a finite sequence has a last term. A formula that defines the nth term of a sequence is called an explicit formula. A recursive formula is one where one or more previous terms are used to generate the next term.

Example 1 Write the first six terms of the sequence defined by tn = 4 n + 5. n 1 2 3 4 5 6 tn 9 13 17 21 25 29

Example 2 Write the first six terms of the sequence defined by the following formula: t 1 = -2 and tn = 2 tn-1 – 1, where t 2 = 2 t 1 – 1 = 2(-2) – 1 = -5 t 3 = 2 t 2 – 1 = 2(-5) – 1 = -11 t 4 = 2 t 3 – 1 = 2(-11) – 1 = -23 t 5 = 2 t 4 – 1 = 2(-23) – 1 = -47 t 6 = 2 t 5 – 1 = 2(-47) – 1 = -95

Series A series is an expression that indicates the sum of terms of a sequence. Summation notation, which uses the Greek letter sigma, , is a way to express a series in abbreviated form. “the sum of 2 n for values of n from 1 to 5”

Example 3 Write the terms of each series. Then evaluate. 2 + 4 + 6 + 8 + 10 + 12 = 42

Summation Properties For sequences ak and bk and positive integer n: n n cak = c ak k =1

Example 4 Write the terms of each series. Then evaluate. = 2(1 + 2 + 3 + 4 + 5 + 6) = 2(21) = 42

Practice Write the first four terms of each sequence. 1) tn = -7 n + 3 2) t 1 = 0 tn = tn-1 - 4

Summation Properties For sequences ak and bk and positive integer n: n n (ak + bk) = ak k =1 + n bk k =1

Example 1 Write the terms of each series. Then evaluate. = (2 + 8 + 18 + 32 + 50) – (1 + 2 + 3 + 4 + 5) = (110) – (15) = 95

Summation Formulas For positive integers n: Constant Series n c = nc k =1

Example 2 Write the terms of each series. Then evaluate. = (4)(3) = 12

Summation Formulas For positive integers n: Constant Series Linear Series n c = nc k =1 n n(n + 1) k= 2 k =1

Example 3 Evaluate. = -30

Example 3 Evaluate. = -650

Summation Formulas For positive integers n: Constant Series Linear Series Quadratic Series n c = nc k =1 n n(n + 1) k= 2 k =1 n k =1 k 2 n(n + 1)(2 n + 1) = 6

Example 4 Evaluate. = 660

Example 5 Evaluate 5 k =1 (7 k 2 – 2 k + 5). 5 5 k =1 = 7 k 2 - 2 k + 7 ( 5 6 11 6 ) – 2( 5 5 k =1 5 6 2 ) + 5 5 = 380

Write the first five terms of each sequence. 1) tn = 6 n - 7 2) a 1 = -1; an = 3 an-1 + 5 Evaluate the sum. 3)

Arithmetic Sequences An arithmetic sequence is a sequence whose successive terms differ by the same number, d, called the common difference.

nth Term of an Arithmetic Sequence tn = t 1 + (n – 1)d tn: nth term t 1: first term d: common difference

Example 1 Find the 10 th term of the sequence defined by t 1 = 7 and tn = tn-1 + 6 tn = t 1 + (n – 1)d t 10 = 7 + (10 – 1)6 t 10 = 61

Example 2 Find the 15 th term of the arithmetic sequence in which t 5 = 7 and t 10 = 22. 1. Find the common difference n 5 6 7 8 9 10 t(n) 7 10 13 16 19 22 d d d 7 + 5 d = 22 5 d = 15 d=3 d d

Example 2 Find the 15 th term of the arithmetic sequence in which t 5 = 7 and t 10 = 22. 2. Find t 1 tn = t 1 + (n – 1)d 7 = t 1 + (5 – 1)3 7 = t 1 + 12 -5 = t 1

Example 2 Find the 15 th term of the arithmetic sequence in which t 5 = 7 and t 10 = 22. 3. Find the 15 th term. tn = t 1 + (n – 1)d t 15 = -5 + (15 – 1)3 t 15 = -5 + 42 t 15 = 37

Example 3 Find the five arithmetic means between 6 and 60. 15 24 33 42 51 6, ___, ___, 60 tn = t 1 + (n – 1)d 60 = 6 + (7 – 1)d 54 = 6 d 9=d

Exercises Find the sum of each series. 1) 2) 3) 4) Find the 12 th term of the sequence defined by t 2 = 6 and t 8 = 24.

Sum of the First n Terms of an Arithmetic Series Sn = n ( t 1 + t n 2 )

Example 1 Given 16, 12, 8, 4, …, find S 11. Sn = n tn = t 1 + (n – 1)d ( t 1 + t n 2 t 11 = 16 + (11 – 1)(-4) t 11 = -24 ) S 11 = 11 ( 16 -24 2 S 11 = 11(-4) S 11 = -44 )

Example 2 Evaluate 21 (5 + 4 k). k =1 t 1 = 9 and t 2 = 13, so d = 4 tn = t 1 + (n – 1)d t 21 = 9 + (21 – 1)4 t 21 = 89 ( ) 9 + 89 = 21 ( ) 2 Sn = n S 21 t 1 + t n 2 S 21 = 21(49) S 21 = 1029

Practice Evaluate 15 (22 – 7 k). k =1