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:

[latex]\begin{align}&{x}_{1}=3 \\ &{x}_{n}=2{x}_{n - 1}-1, \text{ for } n\ge 2 \end{align}[/latex]
We can find the subsequent terms of the recursive formula using the first term.
[latex]\begin{align}&{x}_{1}=3\\ &{x}_{2}=2{x}_{1}-1=2\left(3\right)-1=5\\ &{x}_{3}=2{x}_{2}-1=2\left(5\right)-1=9\\ &{x}_{4}=2{x}_{3}-1=2\left(9\right)-1=17\end{align}[/latex]

So, the first four terms are [latex]3,5,9,\text{ and},17[/latex].

Example: Evaluating a Recursive Formula

Write the first five terms defined by the recursive formula.

[latex]\begin{align}&{x}_{1}=9\\ &{x}_{n}=3{x}_{n - 1}-20\text{, for }n\ge 2\end{align}[/latex]

Try It