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:

\begin{aligned} x_{1} = 3 \\ x_{n} = 2 x_{n - 1} - 1, for n \geq 2 \end{aligned}

$\begin{align}&{x}_{1}=3 \\ &{x}_{n}=2{x}_{n - 1}-1, \text{ for } n\ge 2 \end{align}$

We can find the subsequent terms of the recursive formula using the first term.

\begin{aligned} x_{1} = 3 \\ x_{2} = 2 x_{1} - 1 = 2 (3) - 1 = 5 \\ x_{3} = 2 x_{2} - 1 = 2 (5) - 1 = 9 \\ x_{4} = 2 x_{3} - 1 = 2 (9) - 1 = 17 \end{aligned}

$\begin{align}&{x}_{1}=3\\ &{x}_{2}=2{x}_{1}-1=2\left(3\right)-1=5\\ &{x}_{3}=2{x}_{2}-1=2\left(5\right)-1=9\\ &{x}_{4}=2{x}_{3}-1=2\left(9\right)-1=17\end{align}$

So, the first four terms are $3,5,9,\text{ and},17$ .

Example: Evaluating a Recursive Formula

Write the first five terms defined by the recursive formula.

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

Show Solution

The first term is given in the formula. For each subsequent term, we replace ${x}_{n - 1}$ with the value of the preceding term.

$\begin{align}&n=1&& {x}_{1}=9 \\ &n=2&& {x}_{2}=3{x}_{1}-20=3\left(9\right)-20=27 - 20=7 \\ &n=3&& {x}_{3}=3{x}_{2}-20=3\left(7\right)-20=21 - 20=1 \\ &n=4&& {x}_{4}=3{x}_{3}-20=3\left(1\right)-20=3 - 20=-17 \\ &n=5&& {x}_{5}=3{x}_{4}-20=3\left(-17\right)-20=-51 - 20=-71\end{align}$

The first five terms are $9, 7, 1, -17, \text{ and}, -71$ .

Applications of Derivatives: Prerequisite Material