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Iterative Formula Calculator

General form for iterative processes to find approximate solutions.

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Next Term (x₋₊₁)

Formula first

Overview

An iterative formula defines a sequence where each subsequent term is calculated by applying a specific function to the previous term. This recursive process is primarily used in numerical analysis to find approximate solutions to equations that cannot be solved through standard algebraic methods.

Symbols

Variables

= Current Term (x₋), = Next Term (x₋₊₁)

Current Term (x₋)
Variable
Next Term (x₋₊₁)
Variable

Apply it well

When To Use

When to use: These formulas are used when searching for roots of functions or solving non-linear equations where an exact analytical solution is difficult to obtain. They are particularly effective when an initial estimate of the solution is known, allowing for successive refinement.

Why it matters: Iteration is the backbone of modern computational science, enabling computers to solve complex engineering problems and simulate physical systems. From predicting weather patterns to optimizing financial models, iterative algorithms provide the necessary precision for high-level decision making.

Avoid these traps

Common Mistakes

  • Incorrect substitution in complex formulas.
  • Convert units and scales before substituting, especially percentages, time units, or powers of ten.
  • Interpret the answer with its unit and context; a percentage, rate, ratio, and physical quantity do not mean the same thing.

One free problem

Practice Problem

Practice Problem 1

An iterative formula is given by xn1 = √(xn + 6). If the current term xn is 3, calculate the value of the next term xn1.

Current Term (x₋)3

Solve for: xn1

Hint: Substitute the value of xn into the expression under the square root and solve.

Practice Problem 2

Given the iterative relationship xn1 = 2 + (1 / xn), find the value of xn1 when xn is 4.

Current Term (x₋)4

Solve for: xn1

Hint: First calculate the reciprocal of xn, then add 2 to the result.

Practice Problem 3

A sequence is defined by the iteration xn1 = (xn³ + 10) / 9. If xn = 2, calculate the value of the next term xn1.

Current Term (x₋)2

Solve for: xn1

Hint: Cube the value of xn, add 10, and then divide the entire sum by 9.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Wikipedia: Iterative method
  2. Wikipedia: Fixed-point iteration
  3. Numerical Analysis by Richard L. Burden and J. Douglas Faires
  4. Higher Engineering Mathematics by John Bird
  5. Richard L. Burden and J. Douglas Faires, Numerical Analysis
  6. Wikipedia: Newton's method
  7. AQA GCSE Maths — Algebra (Iterative Methods)