Geometric sequence of 20 terms - calculate sum

    Model exponential growth, decay, and compound patterns. Enter the first term, common ratio, and position to get the nth term and partial sum of a geometric sequence.

    A geometric sequence with 20 terms is a sequence where each term is multiplied by a constant ratio (q). The sum of 20 terms for q not equal to 1 is S = a1 x (1 - q^n) / (1 - q). Enter the first term and ratio, and the calculator will list all 20 terms with their sum.

    Parameters

    Enter data for calculations

    The starting value of the sequence

    Each term is multiplied by this ratio

    Which term to calculate

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    Doubling, tripling, halving - how multiplicative patterns work

    A geometric sequence multiplies each term by a fixed ratio r. When r > 1, the sequence explodes upward. When 0 < r < 1, it decays toward zero. The calculator takes the first term a1, ratio r, and position n, then returns the nth term (a1 x rn-1) and the partial sum using the closed-form formula.

    Try it
    a1 = 3, r = 2, n = 8. The sequence doubles each step: 3, 6, 12, 24, 48, 96, 192, 384. Term a8 = 384. Sum S8 = 765.

    Core formulas

    Four equations cover every geometric sequence problem:

    Nth term an = a1 x rn-1
    Partial sum (r != 1) Sn = a1(1 - rn) / (1 - r)
    Partial sum (r = 1) Sn = n x a1
    Infinite sum (|r| < 1) S = a1 / (1 - r)

    Growth behavior by ratio

    Ratio r Behavior Example (a1 = 100)
    r = 2 Doubling (exponential growth) 100, 200, 400, 800, 1600...
    r = 1.08 8% compound growth 100, 108, 116.64, 125.97...
    r = 0.5 Halving (exponential decay) 100, 50, 25, 12.5, 6.25...
    r = -1 Alternating signs 100, -100, 100, -100...
    r = 1 Constant (degenerate case) 100, 100, 100, 100...

    Practical examples

    Compound interest: $10,000 at 8% annual return. a1 = 10000, r = 1.08, n = 20 years. a20 = 10000 x 1.0819 = $43,157. Your money quadruples in 19 years.
    Compound interest is a geometric sequence with r = 1 + rate
    Radioactive half-life: 1000 atoms with half-life decay. a1 = 1000, r = 0.5, n = 10. a10 = 1000 x 0.59 = 1.95 atoms. After 10 half-lives, less than 2 atoms remain.
    Half-life is r = 0.5 applied repeatedly
    Viral spread: 1 person infects 3 others each cycle. a1 = 1, r = 3, n = 10. a10 = 39 = 19,683 new infections in cycle 10. Total S10 = 29,524.
    R0 in epidemiology is the common ratio
    Paper folding: A sheet 0.1 mm thick, folded in half n times. a1 = 0.1, r = 2, n = 42. a42 = 0.1 x 241 = 219,902 km - past the Moon.
    42 folds reaches the Moon - exponential growth is counterintuitive
    Depreciation: A car loses 15% of value each year. a1 = $30,000, r = 0.85, n = 5. a5 = 30000 x 0.854 = $15,660.
    Declining-balance depreciation is geometric with r = 1 - rate

    FAQ

    What is the difference between arithmetic and geometric sequences?
    Arithmetic adds a constant (d) each step - linear growth. Geometric multiplies by a constant (r) each step - exponential growth. A salary that increases by $500/year is arithmetic. An investment growing 8%/year is geometric. After enough terms, geometric always overtakes arithmetic (for r > 1).
    What happens when r is between 0 and 1?
    The sequence decays toward zero. Each term is a fraction of the previous one. With r = 0.5, the sequence halves each step. The infinite sum converges to a1 / (1 - r). For example, 1 + 0.5 + 0.25 + ... = 1/(1 - 0.5) = 2.
    Can r be negative?
    Yes. A negative ratio makes the terms alternate in sign. With a1 = 5, r = -2: the sequence is 5, -10, 20, -40, 80... The sum formula still works.
    How is compound interest a geometric sequence?
    If you invest P at annual rate i, your balance after n years is P x (1 + i)n. This is an with a1 = P and r = (1 + i). At 8% interest, r = 1.08. The Rule of 72 says money doubles in roughly 72/i years.
    What is the sum of an infinite geometric series?
    When |r| < 1, the sum converges: S = a1 / (1 - r). For a1 = 10, r = 0.9: S = 10 / 0.1 = 100. When |r| >= 1, the series diverges - no finite sum exists.
    Why does the rice-on-chessboard problem produce such huge numbers?
    1 grain on square 1, doubled each square. a64 = 263 = 9.2 x 1018 grains. Total S64 = 264 - 1 = 18.4 quintillion. More rice than humanity has ever produced.

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    See also

    Geometric sequence with 5 terms - sum and ratioGeometric sequence with 8 terms - sum and ratioGeometric sequence with 10 terms - sum and ratioGeometric sequence with 12 terms - sum and ratioGeometric sequence with 15 terms - sum and ratio

    Calculator verified by the LiczGrupa.pl team

    Content, formulas and results have been reviewed for accuracy and relevance by our team of specialists.

    Natalia Skrzek

    Reviewed by: Natalia Skrzek

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