Recursive 11­4.notebook
November 04, 2015
Recursive Formulas for Sequences
Student Outcomes: Students write sequences with
recursive and explicit formulas.
An EXPLICIT formula allows direct computation of any
term for a sequence a1, a2 , a3 ,...an ,...
Explicit means just state a rule that works
if you plug in the number of the term you want,
rather than an actual term.
For Arithmetic it is
Geometric it is
an = a1 + d(n-1)
an = a1 r
.
(n - 1)
Recursive 11­4.notebook
November 04, 2015
For each of the following
a) State Geometric or arithmetic
b) Write the explicit formula
c) Find the 10th term
­4, ­28, ­196, ­1372,...
­3.7, ­2.4, ­1.1, 0.2,...
Recursion is the process of choosing a starting term and
repeatedly applying the same process to each term to
arrive at the following term.
A recursive formula always has two parts:
1. the starting value for a1. WHICH MUST BE STATED!
2. the recursion equation for an as a function of an-1 (the
term before it.)
.
For Arithmetic it is: an = an-1 + d and a1 = #
Geometric it is:
an = an-1 r and a1 = #
.
Recursive 11­4.notebook
November 04, 2015
Given the recursive formula for the geometric sequence, find the common ratio, the first five terms and the explicit formula.
.
an = an-1 5
a1 = 2
.
an = an-1 3
a1 = -3
Given the recursive formula for the arithmetic sequence, find the common difference, the first five terms and the explicit formula.
an = an-1 + d
Arithmetic it is
an = an-1 - 5
a1 = 39
an = an-1 - 13
a1 = 3
5
Recursive 11­4.notebook
November 04, 2015
Given a term in the arithmetic sequence and the common difference, find the recursive formula and the next three terms in the sequence.
an = an-1 + d
Arithmetic it is
a7 = 27, d = 13
a23 = 28.6, d = 1.8
Given the first term and the common ratio of a geometric sequence, find the recursive formula and the next three terms in the sequence.
Geometric it is
an = an-1 r
.
a1 = -4, r = 6
­4, ­24, ­144, ­864, ­5184 .
.
an = an-1 r
an = an-1 6
a1 = 40, r = 0.5
Recursive 11­4.notebook
November 04, 2015
Given a term in a geometric sequence and the common ratio, find the first five terms, the explicit formula and the recursive formula. a4 = 24, r = 6