Recursive Formulas for Sequences

New York State Common Core Math Module 3, Algebra I, Lesson 2

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Lesson 2 Summary

Recursive Sequence: An example of a recursive sequence is a sequence that
(1) is defined by specifying the values of one or more initial terms and
(2) has the property that the remaining terms satisfy a recursive formula that describes the value of a term based upon an expression in numbers, previous terms, or the index of the term.

An explicit formula specifies the nth term of a sequence as an expression in n.

A recursive formula specifies the nth term of a sequence as an expression in the previous term (or previous couple of terms).

Exercise

2. Ben made up a recursive formula and used it to generate a sequence. He used B(n) to stand for the nth term of his recursive sequence.

a. What does B(3) mean?
b. What does B(m) mean?
c. If B(n + 1) = 33 and B(n) = 28 , write a possible recursive formula involving B(n + 1) and B(n) that would generate 28 and 33 in the sequence.
d. What does 2B(7) + 6 mean?
e. What does B(n) + B(m) mean?
f. Would it necessarily be the same as B(n + m)?
g. What does B(17) - B(16) mean?

For each sequence, write either a recursive formula.
a) 1, -1, 1, -1, 1, -1, ...
b) 12, 23, 34, 45, ...

4. For each sequence below, an explicit formula is given. Write the first 5 terms of each sequence. Then, write a recursive formula for the sequence.

a) a_n = 2n + 10 for n ≥ 1
b) a_n = (1/2)^n-1 for n ≥ 1