The Fibonacci sequence is one of the simplest patterns in mathematics, and it appears in nature, art, and computing. The rule fits in one sentence: start with 0 and 1, then keep adding the previous two numbers to get the next one. That is all there is to it, but working through the first terms by hand is the fastest way to really understand it.
The rule, stated plainly
Mathematicians write the rule as F(n) = F(n-1) + F(n-2), with F(0) = 0 and F(1) = 1. In plain words: every term after the first two is the sum of the two terms before it. Some books start with 1 and 1 instead of 0 and 1. That gives the same sequence shifted by one position: 1, 1, 2, 3, 5, 8, and so on. Either starting point is fine, but say which one you are using, because the numbering of terms changes with it.
Worked example: the first ten terms
Start with 0 and 1. Add them: 0 + 1 = 1, so the third term is 1. Now add the last two terms, 1 + 1 = 2. Continue: 1 + 2 = 3, then 2 + 3 = 5, then 3 + 5 = 8, then 5 + 8 = 13, then 8 + 13 = 21, then 13 + 21 = 34. The first ten terms are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34. Notice the pattern that makes checking easy: every term is just the sum of the two numbers sitting right before it. If one term looks wrong, check the two before it, because a single early error carries forward into every later term.
How to generate a longer sequence by hand
Write the terms in a single row and keep your place with a finger or a ruler, because it is easy to add the wrong pair once the numbers grow. Work in small blocks of five terms, then pause and re-add the last pair as a check. The numbers grow fast: by the twentieth term you are past 4,000, and by the fortieth past 100 million, so a calculator becomes sensible after about fifteen terms. If you need a specific term, such as the tenth, count carefully from the start and remember whether your list begins at term zero or term one; the tenth term is 55 when counting from F(1) = 1, and 34 when counting from F(0) = 0 in the zero-based list above.
The golden ratio connection
Divide each term by the previous one as you go further along: 8 divided by 5 is 1.6, 13 divided by 8 is 1.625, 21 divided by 13 is about 1.615. The ratios settle closer and closer to about 1.618, a number called the golden ratio. You need ten to fifteen terms before this settling becomes clear, which is a good reason to generate a longer list rather than stopping at five terms. The golden ratio shows up in sunflower spirals, pinecones, and design proportions, which is why the sequence keeps appearing outside mathematics textbooks.
Common mistakes
The most common mistake is adding a term to itself instead of adding the previous two different terms. Another is starting with 1, 2 instead of 0, 1 or 1, 1, which produces a different sequence entirely. A third is losing count of terms, especially when the question asks for the nth term. Label each term with its position as you write it, and your answer will match the question’s numbering.
When to use a generator
For homework up to about fifteen terms, working by hand teaches the rule best. For longer lists, classroom demonstrations, or checking your hand-written work, a generator produces the sequence instantly and removes arithmetic slips. Generate the list, then verify the first few terms against your hand calculation; if they match, you can trust the rest.
Try it free right here
Use the free Fibonacci Sequence Generator Tool below, right on this page. It runs in your browser and needs no sign-up. You can also open the full tool here: Fibonacci Sequence Generator Tool.