Suffix Notation

## Suffix Notation

A suffix notation is a way of identifying different versions of the same variable. This is often used when repeating the same steps many times, a process called iteration.

In step 1 of an iteration, a variable might be shown as x_1, with the small 1 (the suffix) indicating that that is the value of x in step 1. If the value of x changes in step 2, then the new value is shown as x_2.

An initial value for x would be shown as x_0.

Some processes are recursive: that is, the answer from one step can be used as the input to the next step, with each step in the process taking you closer to the answer. Typically, the formula would be shown as x_(n+1)=root(3)(20 - x_n^2 - x_n).

## Example 1

Carry out the iterative formula x_(n+1)=root(3)(x_n^3 - x_n) with a starting value of x_0 for 5 iterations to obtain an approximate solution to the formula x^3 + 2x = 30.

Start with a value of x_0 = 2.

 step 1 2 root(3)(30-2(2)) 2.9625 step 2 2.9635 root(3)(30-2(2.962496)) 2.8875 step 3 2.8875 root(3)(30-2(2.887501)) 2.89349 step 4 2.89349 root(3)(30-2(2.893485)) 2.89301 step 5 2.89301 root(3)(30-2(2.893009)) 2.89305

## Example 2

By using an iterative method, or otherwise, solve the equation x^3 + 2x = 40 to three decimal places.

Rearrange the formula

x^3 + 2x = 40

x^3 = 40 - 2x

x = root(3)(40 - 2x)

The iterative formula is x_(n+1) = root(3)(40 - 2_n)

 step 1 2 root(3)(40 - 2(2)) 3.30193 3.302 (3dp) step 2 3.30193 root(3)(40 - 2(3.301927)) 3.22032 3.220 (3dp) step 3 2.22032 root(3)(40 - 2(3.220318)) 3.22556 3.226 (3dp) step 4 3.22556 root(3)(40 - 2(3.225556)) 3.22522 3.225 (3dp) step 5 3.22522 root(3)(40 - 2(3.225220)) 3.22524 3.225 (3dp)