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The Sequence Calculator generates terms and computes partial sums for three fundamental types of mathematical sequences: arithmetic, geometric, and Fibonacci-type. Sequences are ordered lists of numbers that follow a specific pattern, and they are foundational to virtually every area of mathematics, from calculus and number theory to computer science and financial modeling.
An arithmetic sequence has a constant difference $$d$$ between consecutive terms: $$a, a+d, a+2d, \ldots$$. The $$n$$-th term is $$a_n = a_1 + (n-1)d$$ and the sum of the first $$n$$ terms is $$S_n = \frac{n}{2}(2a_1 + (n-1)d)$$. Arithmetic sequences appear in evenly spaced measurements, linear depreciation, and arithmetic progressions in number theory. Gauss famously computed $$1 + 2 + \cdots + 100 = 5050$$ at age seven using the arithmetic sum formula.
A geometric sequence has a constant ratio $$r$$ between consecutive terms: $$a, ar, ar^2, \ldots$$. The $$n$$-th term is $$a_n = a_1 r^{n-1}$$ and the partial sum is $$S_n = a_1 \frac{1 - r^n}{1 - r}$$ for $$r \neq 1$$. Geometric sequences model exponential growth, compound interest, geometric decay, and fractal patterns. When $$|r| < 1$$, the infinite series converges to $$S_{\infty} = a_1/(1-r)$$, forming the basis for present value calculations in finance.
A Fibonacci-type sequence defines each term as the sum of the two preceding terms: $$F_1, F_2, F_1+F_2, \ldots$$. The classic Fibonacci sequence uses $$F_1 = 1, F_2 = 1$$, but this calculator allows any starting pair. Fibonacci numbers appear throughout nature — in spiral arrangements of sunflower seeds, the branching of trees, the shell of the nautilus, and the proportions of the human body. The ratio of consecutive Fibonacci numbers approaches the golden ratio $$\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618$$.
Select a sequence type, enter the starting parameters, and choose how many terms to display (up to 10). The calculator generates each term individually and computes the partial sum, giving you both the pattern and the accumulated total.
Arithmetic Sequence: Given first term $$a_1$$ and common difference $$d$$:
$$a_n = a_1 + (n-1)d$$
$$S_n = \frac{n}{2}(2a_1 + (n-1)d) = \frac{n(a_1 + a_n)}{2}$$
Geometric Sequence: Given first term $$a_1$$ and common ratio $$r$$:
$$a_n = a_1 \cdot r^{n-1}$$
$$S_n = a_1 \cdot \frac{1 - r^n}{1 - r} \quad (r \neq 1)$$
When $$r = 1$$, all terms equal $$a_1$$ and $$S_n = na_1$$.
Fibonacci-type Sequence: Given $$F_1$$ and $$F_2$$:
$$F_n = F_{n-1} + F_{n-2} \quad \text{for } n \geq 3$$
The sum is computed by direct addition of the generated terms. There is no simple closed-form for the partial sum of a generalized Fibonacci sequence, though Binet's formula gives a closed form for individual terms of the standard Fibonacci sequence.
Terms 1-10 display the individual sequence values. For arithmetic sequences, the differences between consecutive terms are constant. For geometric sequences, the ratios are constant. For Fibonacci-type, each term equals the sum of the two before it.
Sum of First n Terms is the partial sum $$S_n$$. For arithmetic sequences, this grows quadratically in $$n$$. For geometric sequences with $$|r| > 1$$, the sum grows exponentially. For geometric sequences with $$|r| < 1$$, the sum converges to a finite limit. For Fibonacci-type sequences, the sum grows exponentially (at a rate approaching $$\phi$$).
Terms beyond the selected $$n$$ display as zero — these are not part of the sequence, just unused output slots.
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The sequence is 2, 7, 12, 17, 22, 27 with constant difference d = 5. Sum S₆ = 6(2+27)/2 = 6·29/2 = 87. Each term increases by exactly 5.
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The classic Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. Sum = 143. Note the ratio of consecutive terms approaches φ ≈ 1.618: 55/34 ≈ 1.6176.
A sequence is an ordered list of numbers (1, 3, 5, 7, ...). A series is the sum of a sequence's terms (1 + 3 + 5 + 7 + ...). This calculator shows both: individual terms (the sequence) and the partial sum (the series). The distinction matters in convergence analysis — a sequence may converge while its series diverges.
Fibonacci numbers appear throughout nature, art, and mathematics. They describe spiral patterns in plants, branching in trees, and proportions in architecture. In mathematics, they connect to the golden ratio, combinatorics, and number theory. In computer science, they arise in algorithm analysis (Fibonacci heaps, optimal search).
When r = -1, the terms alternate: a, -a, a, -a, ... The partial sums oscillate between a and 0. This sequence diverges (does not approach a limit), though its Cesàro sum is a/2. Such alternating sequences appear in signal processing and Fourier analysis.
Yes. Common difference d and ratio r can be any real number, including fractions and decimals. For example, d = 0.5 gives the sequence 1, 1.5, 2, 2.5, ... and r = 0.5 gives 1, 0.5, 0.25, 0.125, ... (a converging geometric series).
This calculator displays up to 10 terms. For practical purposes, 10 terms are sufficient to identify the pattern and compute partial sums. The closed-form formulas (arithmetic and geometric) can evaluate any specific term aₙ for arbitrarily large n without generating all intermediate terms.
Compound interest creates a geometric sequence. If you invest P dollars at annual rate r, your balance after n years is P·(1+r)ⁿ — a geometric sequence with ratio (1+r). The partial sum formula gives the total value of regular deposits (annuity formula).
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The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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