Fibonacci Sequence Calculator NZ

This calculator generates the Fibonacci sequence and finds any term, along with the running sum and the famous golden ratio the sequence approaches. The Fibonacci sequence is one of the most celebrated patterns in mathematics: each number is the sum of the two before it, starting from 0 and 1, giving 0, 1, 1, 2, 3, 5, 8, 13, 21 and on. It appears throughout nature, in the spiral of shells and sunflowers, the branching of plants and the breeding of rabbits in the original 13th-century problem, and it underpins ideas in art, architecture, computer science and financial analysis. A striking property is that the ratio of consecutive terms gets closer and closer to the golden ratio, about 1.618, a number prized for its aesthetic proportions. You simply enter how many terms you want, and the calculator lists the sequence, gives the value of the final, nth term, the sum of all the terms generated, and the ratio of the last two terms, which converges on the golden ratio. The results update as you change the number of terms, so you can watch the ratio settle toward 1.618 and see how quickly the numbers grow. Use it for homework, to explore the pattern, to demonstrate the golden ratio, or as a reference when programming or designing. The sequence grows very fast, so this tool caps the number of terms to keep the values exact and readable. The calculations are exact integers up to that cap.

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89
value of the nth term
Sum of terms232
Golden ratio approx1.618
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Each term is the sum of the two before it, starting 0, 1. The ratio of consecutive terms approaches the golden ratio (~1.618). Capped at 40 terms.

How it works

The sequence starts with 0 and 1, and each following term is the sum of the previous two. The nth term is the last one generated. The sum adds all the terms listed. The golden ratio approximation is the last term divided by the one before it, which converges to about 1.618 as the sequence grows.

Worked example

With 12 terms, the sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89. The twelfth term is 89, the sum of all twelve is 232, and the ratio 89 divided by 55 is about 1.618, the golden ratio. Each step, the ratio of neighbouring terms gets closer to that famous number.

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Frequently asked questions

What is the Fibonacci sequence?

It is a series where each number is the sum of the two before it, starting from 0 and 1: 0, 1, 1, 2, 3, 5, 8, 13 and so on. It appears throughout nature and mathematics and is closely linked to the golden ratio.

How is the Fibonacci sequence related to the golden ratio?

As the sequence grows, the ratio of each term to the one before it gets closer and closer to the golden ratio, approximately 1.618. This calculator shows that ratio so you can watch it converge as you add more terms.

Why is the number of terms capped?

Fibonacci numbers grow exponentially and quickly exceed the range where ordinary arithmetic stays exact. Capping the number of terms keeps every value precise and the output readable, while still showing the pattern and the golden-ratio convergence clearly.