Algebra 1 Common Core Student Edition, Grade 8-9

Writing a Recursive Formula

Write a recursive formula for the arithmetic sequence below. What is the value of the

8th term?

+7 +7

Step 1 A(l) = 70 First te rm o f th e sequence

A{2) = 71(1) + 7 = 70 + 7 = 77 A(2) is fo u n d by a dd ing 7 to >A( 1).

>4(3) = >4(2) + 7 = 77 + 7 = 84 A(3) is fo u n d by a dd ing 7 to A{2).

>4(4) = >4(3) + 7 = 84 + 7 = 91 >A(4) is fo u n d by a dd ing 7 to >4(3).

>4(n) = A{n - 1) + 7 A(r i ) is fo u n d by a dd ing 7 to A{n - 1).

I h e recursive form ula for th e arithm etic sequence is A{n) = A(n - 1) + 7, where

>4(1) = 70.

Step 2 To find the value of th e 8th term , you n e e d to extend th e pattern.

>4(5) = >4(4) + 7 = 91 + 7 = 98

>4(6) = >4(5) + 7 = 98 + 7 = 105

>4(7) = >4(6) + 7 = 105 + 7 = 112

>4(8) = >4(7) + 7 = 112 + 7 = 119

The value of th e 8th term is 119.

© & G o t It? 3. Write a recursive form ula for each arithm etic sequence. W hat is the 9th

term of each sequence?

a. 3, 9 , 1 5 , 2 1 ,... b. 23, 3 5 ,4 7 , 5 9 ,...

C. 7.3, 7.8, 8.3, 8. 8 ,. .. d. 97, 88, 79, 7 0 ,...

e. Reasoning Is a recursive form ula a useful way to find th e value of an

arithm etic sequence? Explain.

You can find the value of any term of an arithm etic seq u en ce using a recursive formula.

You can also w rite a sequence using an explicit formula. An explicit form ula is a

function rule th a t relates each term of a seq u e n ce to th e term num ber.

Key Concept Explicit Formula For an Arithmetic Sequence

The nth term of an arithm etic sequence w ith first term >4(1) an d co m m o n difference d is

given by

>4(n) = >4(1) + ( n - 1 )d

f f f T\

n th term first term term n u m b e r c o m m o n difference

276 Chap t er 4 An Int ro d uct io n t o Funct ions