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Explanation of a Recursive Formula
A recursive function contains two important components:
1.A recursion formula that tells how any term of a sequence relates to the
preceding terms (e.g., xn = xn - 1 + 2, or Next = Now + 2); and
2.An initial condition that gives the starting point (e.g., x1 = 1 or Start =
The initial condition is necessary to ensure a uniquely defined sequence. The
example above gives the sequence of odd numbers 1, 3, 5, 7, ... . However,
if the initial condition was modified to x1 = 2 or Start = 2, the recursive
function would give the sequence of even numbers 2, 4, 6, 8, ... .
Unlike a recursive formula, an explicit formula stands alone; that is, it has no
additional qualifiers. The explicit formula y = x2 can be used to determine the
xth square number when you know the value of y. In contrast, the recursive
formula for the square numbers is an = an - 1 + 2n - 1 where a1 = 1. To find
the nth square number, you first need to find the previous n - 1 square
Other examples:
• The linear function y = 60x + 230, which describes the line with y-intercept
230 and slope 60, can be used to represent the distance from home
(y) of a traveler who began the day 230 miles from home and drives
at 60 miles per hour for x hours. Defined recursively, this relationship
would be Next = Now + 60 where Start = 230. •
The balance of a bank account that earns 3% a year can be defined
recursively as Next = Now + 0.03 * Now or as Next = 1.03 * Now. If
the beginning balance in the account was $248, then Start= 248.