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Fibonacci Sum Calculator

Add up the first n Fibonacci numbers and see the nth term at the same time.

How it is calculated

Enter the number of terms n, a whole number. The tool generates the sequence starting F₁ = 1, F₂ = 1, F₃ = 2, … and adds the first n terms, then reports the sum and the nth term itself. The sum satisfies F₁ + … + Fₙ = Fₙ₊₂ − 1. The arithmetic uses exact 64-bit integers, so every digit shown is correct. That limits n to 90: from 91 terms the sum no longer fits in a 64-bit integer, and the tool shows no result rather than a rounded number. A decimal such as 7.5, zero, negatives and anything above 1000 also give no result. At n = 90 the sum is 7,540,113,804,746,346,428.

Worked example

For n = 10 the first ten terms are 1, 1, 2, 3, 5, 8, 13, 21, 34 and 55. Adding them: 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143. The 10th term is 55. The identity checks out: the 12th Fibonacci number is 144, and 144 − 1 = 143. This is why the sum of the first ten terms always sits one below a later term.

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Common mistakes

  • Assuming the sequence starts at 0. Here the first term is 1, so the 10th term is 55; counting from F₀ = 0 shifts every result by one place.
  • Confusing the sum with the nth term. For n = 10, the sum is 143 but the 10th term is only 55.
  • Entering 100 or 200 expecting a big number. Above 90 terms the exact value is too large for the tool and no result is shown.
  • Entering a decimal count such as 7.5, expecting a partial term. Only whole numbers are accepted.

About this calculator

The Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13, 21...) is built by adding the two previous terms together, but the sum of the first n terms follows its own elegant closed-form identity: F₁ + F₂ + ... + Fₙ = Fₙ₊₂ − 1. In other words, adding up any run of consecutive Fibonacci numbers starting from the beginning always lands one short of a later Fibonacci number — a pattern that can be proven neatly by induction and is a staple exercise for building comfort with recursive sequences.

Fibonacci sums and the sequence itself extend well beyond pure math: computer science uses Fibonacci-based structures in algorithm analysis (Fibonacci heaps, dynamic programming exercises) and their sums appear in complexity estimates; botanists and biologists point to Fibonacci-related spirals in sunflower seed heads, pinecones and shell growth as an approximation of how the golden ratio (the sequence's limiting ratio) shows up in efficient natural packing; and financial technical analysts use cumulative Fibonacci-derived levels for retracement and extension analysis in price charts.

This calculator computes both the sum of the first n Fibonacci numbers and the nth Fibonacci term itself in one step, saving you from generating and adding the whole sequence by hand.

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