5.2: Explicit and Recursive Rules | Algebra 2 Honors

How it works

An explicit rule gives a term directly from its position. A recursive rule gives an initial term and a method for obtaining later terms from previous ones. A recursive rule without a starting value generally does not specify a unique sequence.

a_n=a_{n-1}+d\quad(n\ge2),\qquad a_1\text{ specified}

Worked examples

Example 1

Write explicit and recursive rules for $\displaystyle 5,9,13,\ldots$ .

The difference is $\displaystyle 4$ . Explicit: $\displaystyle a_n=5+4(n-1)$ . Recursive: $\displaystyle a_1=5$ , $\displaystyle a_n=a_{n-1}+4$ for $\displaystyle n\ge2$ .

Example 2

Given $\displaystyle a_1=12$ and $\displaystyle a_n=a_{n-1}/2$ , find $\displaystyle a_4$ .

The terms are $\displaystyle 12,6,3,3/2$ , so $\displaystyle a_4=3/2$ .

Problems

Problem 5.2.1

Write a recursive rule for $\displaystyle 2,6,18,\ldots$ .

Show worked solution

$\displaystyle a_1=2$ and $\displaystyle a_n=3a_{n-1}$ for $\displaystyle n\ge2$ .

Problem 5.2.2

Write an explicit rule for $\displaystyle a_1=7$ , $\displaystyle a_n=a_{n-1}-2$ .

Show worked solution

The arithmetic difference is $\displaystyle -2$ , so $\displaystyle a_n=7-2(n-1)$ .

Problem 5.2.3

Given $\displaystyle b_1=3$ , $\displaystyle b_n=2b_{n-1}+1$ , find $\displaystyle b_2,b_3,b_4$ .

Show worked solution

Apply the rule sequentially: $\displaystyle b_2=7$ , $\displaystyle b_3=15$ , and $\displaystyle b_4=31$ .

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 5.2.1

Problem 5.2.2

Problem 5.2.3

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