How Arithmetic, Geometric and Fibonacci Sequences Are Built
A sequence is an ordered list of numbers built by a rule, and three families cover almost everything a student meets. An arithmetic sequence adds the same amount each step, so 1, 3, 5, 7, 9 has a common difference of 2. A geometric sequence multiplies by the same amount each step, so 1, 2, 4, 8, 16 has a common ratio of 2. A Fibonacci sequence adds the two previous terms, giving 0, 1, 1, 2, 3, 5, 8, 13. The rules look simple, but they show up in loan schedules, population models, compound growth, computer algorithms and the branching of plants — which is why exam papers keep asking for the nth term and the sum without listing every value.
For an arithmetic sequence, aₙ = a₁ + (n − 1)d gives the nth term, where a₁ is the first term, d is the common difference and n is the position you want. The sum of the first n terms is Sₙ = n/2 × (2a₁ + (n − 1)d), equivalently n(a₁ + aₙ)/2. For a geometric sequence, aₙ = a₁ × r^(n−1), where r is the common ratio, and Sₙ = a₁(1 − rⁿ) ÷ (1 − r) — but only when r ≠ 1, because r = 1 makes the denominator zero. In that case every term equals a₁ and the sum is n × a₁, handled explicitly here rather than as an error. The infinite sum a₁ ÷ (1 − r) exists only when |r| < 1; for |r| ≥ 1 the series diverges. Fibonacci uses aₙ = aₙ₋₁ + aₙ₋₂, computed iteratively, and its first-n sum follows the identity a₍ₙ₊₂₎ − 1.
Run the examples calculator.net publishes. For 1, 3, 5, 7, 9 with a₁ = 1 and d = 2, the fifth term is a₅ = 1 + 2 × (5 − 1) = 9 and the sum of five terms is 5 × (1 + 9) ÷ 2 = 25, which you can confirm by adding them by hand. Push to n = 10 and you get a₁₀ = 19 with S₁₀ = 100 — the sum of the first ten odd numbers is 10², a result worth remembering. For the geometric sequence 1, 2, 4, 8 with r = 2, the eighth term is 1 × 2⁷ = 128 and the sum of eight terms is 255, one less than 2⁸. Set a₁ = 1 and r = 0.5 and the infinite sum is 1 ÷ 0.5 = 2, while r = 2 diverges. Fibonacci at n = 10 gives 55, and at n = 100 gives 354,224,848,179,261,915,075.
Three uses come up repeatedly. A student revising arithmetic and geometric progressions needs the substituted working, not just the answer, because marks come from showing a₁ + (n − 1)d with real numbers in it. Anyone modelling something that grows by a fixed percentage — a salary rising 8% a year, bacteria doubling every hour — is working with a geometric sequence, and the sum of n terms is the total accumulated over that span. And the pattern-detection mode is genuinely handy when you have a list of terms from a question or a spreadsheet and need to know what kind of sequence it is: paste 2, 5, 8, 11, 14 and it reports arithmetic with d = 3 and predicts the next term.
One precision detail matters here more than anywhere else in basic maths. F(78) = 8,944,394,323,791,464 is the last Fibonacci number below JavaScript's safe integer limit of 9,007,199,254,740,991. Past that point a calculator using ordinary floating-point numbers corrupts the final digits without warning, so F(79) can come back as 14,472,334,024,676,220 rather than the correct 14,472,334,024,676,221. This tool adds terms digit by digit, keeping every one exact, and tells you when you cross that threshold. Two other honest limits: Fibonacci indexing differs between textbooks, so both a₀ = 0 and F₁ = 1 forms are offered with the active one named, and the detector recognises only the three named families. Definitions follow Wolfram MathWorld and OEIS A000045. All computation happens in your browser.
