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# Iteration

## Find approximate solutions to equations numerically using iteration.

##### FibonacciLevel 1Level 2Level 3Level 4Exam-StyleDescriptionHelpMore Algebra

This is level 4: solving equations to one decimal place. You can earn a trophy if you get at least 5 questions correct.

 1. Use the iterative formula $$x_{n+1}= \frac{5}{1+x_n}$$ to find the value of $$x$$ to three significant figures starting with $$x_0=1$$. 2. Use the iterative formula $$x_{n+1}= \ \sqrt{x_n+8}$$ to find the value of $$x$$ to four decimal places starting with $$x_0=3$$. 3. The equation $$x^2-5x-4=0$$ can be arranged to give the iterative formula $$x_{n+1}= \sqrt{5x_n+4}$$. Find the value of $$x$$ to three significant figures obtained by starting with $$x_0=5.5$$. 4. The equation $$x^3+4x=1$$ can be arranged to give the iterative formula $$x_{n+1}= \frac{1-x_n^3}{4}$$. Find the value of $$x$$ to three significant figures obtained by starting with $$x_0=0.5$$. 5. The equation $$x^3-8x-3=0$$ can be arranged to give the iterative formula $$x_{n+1}= \frac{x_n^3-3}{8}$$. Find the value of $$x$$ to three significant figures obtained by starting with $$x_0=-1$$. 6. Use the iterative formula $$x_{n+1}= \sqrt{\frac{x_n(x_n^2+3)}{5}}$$. Find the value of $$x$$ to three decimal places obtained by starting with $$x_0=1$$.
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This is Iteration level 4. You can also try:
Level 1 Level 2 Level 3

## Instructions

Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help.

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## Description of Levels

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Fibonacci Probably the first iterative example learners will encounter is the Finonacci sequence. A simple rule exists for finding the next term of the sequence from the previous two.

Level 1 - Generating sequences using next term rule.

Level 2 - Rearranging equations.

Level 3 - Using flowcharts to define iterations.

Level 4 - Solving equations to one decimal place.

Exam Style questions are in the style of GCSE or IB/A-level exam paper questions and worked solutions are available for Transum subscribers.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

## Example

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly.

For an animated demonstration of the calculator button pressing order for iteration see Calculator Workout skill 16.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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