And work with general iterative processes

Iteration is a general process: you take a starting value, apply one rule, and feed the result back in as the next input. Each output becomes the next input, so the rule repeats again and again. Higher tier work uses this repetition to build sequences and to find approximate solutions to equations. The notation uses subscripts as position labels. If x n plus one equals f of x n, then x zero is the starting value, x one is f of x zero, and x two is f of x one. Read that subscript as a position label, not as an arithmetic operation. To generate terms, substitute the starting value into the formula to get the first term. Then apply the same rule again, using that output as the next input. Write each substitution line clearly, and keep full calculator accuracy, because rounding intermediate values too early can spoil the final converged value. Let's try x n plus one equals three x n minus two, with x zero equal to four. Pause here and predict x one before I show it. Substituting four gives x one equals three times four minus two, which is ten. Now feed ten back in: x two equals three times ten minus two, which is twenty-eight. And again: x three equals three times twenty-eight minus two, which is eighty-two. Three mistakes are common. Restarting from the original starting value each time instead of feeding the previous output back in. Misreading x n plus one as x n plus one, when the subscript is a position label, not an arithmetic operation. And rounding intermediate values too early, so keep full calculator accuracy at every step. Iteration is the repeated application of a single rule, using each output as the next input. Subscripts label positions, and you generate terms by substituting and repeating in the correct order. Continue an iteration until the values converge and you have an approximate solution to the required degree of accuracy. Use the ANS button to feed each term back quickly, and keep the starting value visible in your working.