An arithmetic and geometric sequence and series calculator for students. Calculate the nth term, sum of the first n terms, infinite series, and detect sequence patterns.
Six tabs: arithmetic nth term (U_n = a + (n-1)b), arithmetic sum of n terms (S_n = n/2 x (2a + (n-1)b)), geometric nth term (U_n = a * r^(n-1)), geometric sum of n terms, infinite geometric series (S_infinity = a/(1-r) when |r|<1), and sequence pattern detection from the first 4-6 numbers to identify arithmetic or geometric type.
Calculator information
📋 How to use this calculator
- Pick the tab that fits your task: nth Term Arithmetic, Sum of n Arithmetic Terms, nth Term Geometric, Geometric Sum, Infinite Geometric Series, or Pattern Detection.
- For arithmetic, enter the first term a, common difference d (the gap between terms), and term index n; the tool returns U_n and a step-by-step solution.
- For an arithmetic sum, enter a, d, and the number of terms n; the output is S_n shown both ways (S_n = n/2*(2a+(n-1)d) and S_n = n/2*(a+U_n)).
- For geometric, enter a, common ratio r, and n; r must be nonzero; the output is U_n = a*r^(n-1).
- For an infinite geometric series, enter a and r; the series converges only when |r| < 1, otherwise it diverges and has no finite sum.
- For pattern detection, enter the first 4-6 terms (comma-separated); the tool checks whether the sequence is arithmetic (constant difference) or geometric (constant ratio) and projects the next terms.
🧮 Arithmetic and geometric sequences and series
Arithmetic: U_n = a + (n-1)*d ; S_n = n/2 * (2a + (n-1)d) ; Geometric: U_n = a*r^(n-1) ; S_n = a*(1 - r^n)/(1 - r) for r != 1 ; Infinite geometric: S_inf = a/(1 - r) for |r| < 1
- a = U_1 = first term
- d = common difference = U_n - U_(n-1)
- r = common ratio = U_n / U_(n-1)
- n = term index (n = 1, 2, 3, ...)
- S_n = sum of the first n terms
- S_inf = sum of an infinite geometric series (converges only when -1 < r < 1)
For r = 1 in a geometric series, S_n = n*a (every term equals a). An arithmetic series diverges as n approaches infinity (unless a = 0 and d = 0). A geometric series converges to a/(1-r) only when |r| < 1.
💡 Worked example: Example: simple-interest savings (arithmetic sequence)
Given:- Opening balance $1,000
- Flat interest of $50 per month (simple, not compounded)
- Find the balance at month 12 and total interest over 12 months
Steps:- Identify: an arithmetic sequence with a = 1,000 + 50 = 1,050 (end of month 1) and d = 50.
- 12th term (balance at end of month 12): U_12 = 1,050 + (12-1)*50 = 1,050 + 550 = $1,600.
- Total interest over 12 months: 12 * 50 = $600 (an arithmetic series with a = 50, d = 0).
- Verify with S_n: S_12 = 12/2 * (2*50 + 11*0) = 6 * 100 = $600.
Result: End-of-month-12 balance is $1,600 with total interest of $600 (a hypothetical 60% annual return, not realistic in practice).
❓ Frequently asked questions
What is the difference between a sequence and a series?
A sequence is an ordered list of numbers, e.g., 2, 5, 8, 11, 14, .... A series is the sum of the terms of a sequence, e.g., 2+5+8+11+14 = 40. The nth term is denoted U_n, and the sum of the first n terms is S_n. They are related by U_n = S_n - S_(n-1). Every sequence yields a series, but not every series converges to a finite value.
When does an infinite geometric series have a finite sum?
An infinite geometric series converges (has a finite sum) if and only if |r| < 1, i.e., -1 < r < 1. Its sum is S_inf = a/(1-r). Example: 1 + 1/2 + 1/4 + 1/8 + ... = 1/(1 - 1/2) = 2. If |r| >= 1 the series diverges or oscillates. This is the key idea behind Zeno's paradox of Achilles and the tortoise, which actually has a finite solution.
How do I detect more complex patterns?
Compute successive differences (d_i = U_(i+1) - U_i): constant means arithmetic. Compute successive ratios (r_i = U_(i+1)/U_i): constant means geometric. If neither is constant, check second differences: constant means a quadratic sequence U_n = a*n^2 + b*n + c. Other patterns include Fibonacci (U_n = U_(n-1) + U_(n-2)), higher-order arithmetic, and harmonic sequences. This calculator only detects arithmetic and geometric patterns.
Where do sequences and series show up in real life?
Arithmetic sequences appear in flat-rate installment loans, straight-line depreciation, and tiered sales targets (e.g., 100 units in month 1, 110 in month 2, 120 in month 3). Geometric sequences appear in compound interest, population growth, radioactive decay, annuity formulas, and present-value calculations in finance. Infinite series underpin advanced math, including Taylor series for sin, cos, and e^x.
How do I find U_n when only two terms are known?
Say U_3 = 11 and U_8 = 31. For arithmetic: U_8 - U_3 = 5d = 20, so d = 4. Then a = U_3 - 2d = 11 - 8 = 3, giving U_n = 3 + (n-1)*4 = 4n - 1. For geometric from two terms U_p and U_q: r^(q-p) = U_q/U_p. Example: U_3 = 12 and U_5 = 48 gives r^2 = 4 so r = 2 or r = -2. Verify by computing U_4 and checking consistency.
📚 Sources & references
Last updated: May 11, 2026