Skills Review for Newton’s Method

Learning Outcomes

  • Write the terms of a sequence defined by a recursive formula

In the Newton’s Method section,  a recursive formula will be used to approximate the x-intercepts of functions. Here we will review how a recursive formula works.

Evaluate a Recursive Formula

A recursive formula is a formula that defines its value at a particular input using the result of the previous input(s).

A recursive formula always has two parts: the value of an initial input and an equation defining each term in terms of preceding terms. For example, suppose we know the following:

x1=3xn=2xn−1−1, for n≥2x1=3xn=2xn−1−1, for n≥2
We can find the subsequent terms of the recursive formula using the first term.
x1=3x2=2x1−1=2(3)−1=5x3=2x2−1=2(5)−1=9x4=2x3−1=2(9)−1=17x1=3x2=2x1−1=2(3)−1=5x3=2x2−1=2(5)−1=9x4=2x3−1=2(9)−1=17

So, the first four terms are 3,5,9, and,173,5,9, and,17.

Example: Evaluating a Recursive Formula

Write the first five terms defined by the recursive formula.

x1=9xn=3xn−1−20, for n≥2x1=9xn=3xn−1−20, for n≥2

Try It