What is an arithmetic sequence?

An arithmetic sequence adds a constant difference d to each term. General form: a₁, a₁+d, a₁+2d, ... The nth term: aₙ = a₁ + (n−1)d. Sum of n terms: Sₙ = n/2 × (2a₁ + (n−1)d). Example: 3, 7, 11, 15 has d=4, a₅=3+4×4=19.

What is a geometric sequence?

A geometric sequence multiplies each term by a constant ratio r. General form: a₁, a₁r, a₁r², ... The nth term: aₙ = a₁ × r^(n−1). Sum: Sₙ = a₁(1−rⁿ)/(1−r) for r≠1. Example: 2, 6, 18, 54 has r=3, a₅=2×3⁴=162.

What is the Fibonacci sequence?

The Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... Each term is the sum of the two before it: F(n) = F(n−1) + F(n−2). F(1)=1, F(2)=1. Enter n to get the nth Fibonacci number and the sum of the first n terms.

What does Find Pattern mode do?

Enter a sequence of numbers separated by commas. The calculator attempts to detect if it is arithmetic (constant difference) or geometric (constant ratio). If detected, it shows the formula, next term, and nth term. If not detected, it suggests trying the arithmetic or geometric modes manually.

What is the sum of the first 100 natural numbers?

Sum = n/2 × (first + last) = 100/2 × (1+100) = 50 × 101 = 5050. This is an arithmetic sequence with a₁=1, d=1, n=100. Famously calculated by Gauss as a child. Use the Arithmetic mode with first term=1, difference=1, n=100.

What is an infinite geometric series?

When |r| < 1, the infinite geometric series converges: S∞ = a₁/(1−r). Example: 1 + 0.5 + 0.25 + 0.125 + ... = 1/(1−0.5) = 2. For |r| ≥ 1, the series diverges (no finite sum). The calculator computes partial sums for any number of terms.

How do I find the nth term of a sequence?

For arithmetic: aₙ = a₁ + (n−1)d. Enter first term, common difference, and which term number you want. For geometric: aₙ = a₁ × r^(n−1). Enter first term, common ratio, and term number. For Fibonacci: enter n directly. For unknown patterns, use Find Pattern mode.

Number Sequence Calculator

Four modes: arithmetic sequences (nth term and sum), geometric sequences, Fibonacci numbers, and automatic pattern detection from a list of terms.

First Term (a₁)

Common Diff (d)

Find nth Term

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Guides & Reference

How It Works

Arithmetic sequence — constant differenceLinear patterns, savings plans, salary increments, physics (constant acceleration).

Enter first term (a₁), common difference (d), and which term number (n) you want. The calculator shows: nth term = a₁ + (n−1)×d, partial sum = n/2 × (2a₁ + (n−1)d), and the first several terms as a list. The sum formula equals n/2 × (first+last), which is Gauss's pairing insight. Terms are limited to n ≤ 50 for display.

aₙ = a₁ + (n−1)d | Sₙ = n/2 × (2a₁ + (n−1)d)

a₁=3, d=4, n=10 → a₁₀=39, S₁₀=210

Geometric sequence — constant ratioExponential growth, compound interest, population growth, radioactive decay.

Enter first term (a₁), common ratio (r), and term number (n). Results: nth term = a₁ × r^(n−1), partial sum = a₁ × (1−rⁿ)/(1−r). For r=1: sum = a₁ × n. The first n terms display as a list (n ≤ 50). For |r| < 1, terms decrease toward zero. For |r| > 1, they grow exponentially. Negative r gives alternating signs.

aₙ = a₁ × r^(n−1) | Sₙ = a₁(1−rⁿ)/(1−r)

a₁=2, r=3, n=6 → a₆=486, S₆=728

Fibonacci — sum of previous two termsNature patterns, golden ratio, algorithm analysis, number theory.

Enter n — the calculator shows F(n) and the sum of the first n Fibonacci numbers. The sequence: 1,1,2,3,5,8,13,21,34,55,89,144,... Useful fact: the sum of the first n Fibonacci numbers = F(n+2) − 1. The ratio F(n+1)/F(n) approaches the golden ratio φ = (1+√5)/2 ≈ 1.61803 as n increases.

F(n) = F(n−1) + F(n−2) | F(1)=F(2)=1 | Sum(1..n) = F(n+2)−1

F(10)=55, F(20)=6765, Sum of first 10 = F(12)−1 = 143

Find Pattern — automatic detectionIdentifying the type of an unknown sequence from given terms.

Enter comma-separated terms (e.g. 3,6,12,24,48). The calculator computes consecutive differences (for arithmetic detection) and consecutive ratios (for geometric detection). If all differences are equal → arithmetic, shows d and formula. If all ratios are equal → geometric, shows r and formula. If neither, reports "pattern not detected" and suggests entering manually.

Arithmetic: const diff d | Geometric: const ratio r

Input 3,9,27,81 → geometric r=3 | Input 5,8,11,14 → arithmetic d=3

Partial sums — Sₙ vs S∞Finance, physics, infinite series analysis.

The calculator computes Sₙ for any finite n. For geometric series with |r| < 1: as n grows, Sₙ approaches S∞ = a₁/(1−r). Example: a₁=1, r=0.5: S₁=1, S₅=1.9375, S₁₀=1.998, S∞=2. For arithmetic series, the sum always grows without bound. Compound interest uses geometric sum: total = payment × (rⁿ−1)/(r−1) where r=1+rate.

S∞ = a₁/(1−r) for |r|<1 | Arithmetic Sₙ → ∞ always

a₁=1, r=0.5: S₁₀=1.998, S₂₀=1.999998, S∞=2.000000

Quick Reference

Verify these in the calculator above.

Arithmetic sum

a₁=1, d=1, n=100 (sum)

5050

Arithmetic nth

a₁=1, d=1, n=10 (10th term)

Geometric nth

a₁=2, r=3, n=6 (6th term)

486

Infinite sum

a₁=1, r=0.5, S∞

Fibonacci

Fibonacci F(10)

Fibonacci

Fibonacci F(20)

6,765

Find Pattern

Pattern: 3,6,12,24

Geometric r=2

Find Pattern

Pattern: 5,8,11,14

Arithmetic d=3

Tips & Shortcuts

✓

For the sum of integers 1 to n: use Arithmetic mode with first term=1, common difference=1, and enter your n. Gauss's formula gives Sₙ=n(n+1)/2 instantly.

✓

For compound interest: use Geometric mode with a₁=principal, r=1+rate, and n=number of periods. The nth term gives the balance after n periods.

✓

The Find Pattern mode works best with at least 4-5 terms — fewer terms may not provide enough data to distinguish arithmetic from geometric.

✓

Negative common ratios (r<0) produce alternating positive-negative sequences. Example: a₁=2, r=−3: 2, −6, 18, −54, ...

✓

The golden ratio connection: compute F(20)/F(19) = 6765/4181 ≈ 1.61803 — very close to φ. Verify using the Fibonacci tab.

Common Mistakes

Confusing common difference (arithmetic) with common ratio (geometric)

Arithmetic: consecutive terms differ by a constant (5,8,11 differ by 3). Geometric: consecutive terms have a constant ratio (2,6,18 have ratio 3). Use Find Pattern to detect which type your sequence is.

Entering n=0 or n<1 for the term number

Sequences are indexed from n=1. The first term is a₁ (n=1), the second is a₂ (n=2), etc. The calculator accepts n from 1 to 50 for term display.

Using r=1 in the geometric sum formula

The geometric sum formula Sₙ=a₁(1−rⁿ)/(1−r) is undefined at r=1 (division by zero). For r=1, all terms equal a₁ and Sₙ=n×a₁. The calculator handles this case separately.

Expecting the infinite sum to converge when |r| ≥ 1

S∞ = a₁/(1−r) only works for |r| < 1. If r=1.05 (compound interest), the sum grows without bound — there is no finite infinite sum. The partial sum formula Sₙ still applies for any finite n.

Confusing the Fibonacci sequence start with F(0)=0,F(1)=1

This calculator uses F(1)=1, F(2)=1 (1-indexed). Some definitions start F(0)=0. Verify: F(1)=1, F(2)=1, F(3)=2, F(4)=3, F(5)=5 matches the expected sequence for your problem.

Frequently Asked Questions

An arithmetic sequence adds a fixed common difference d to each term: a₁, a₁+d, a₁+2d, a₁+3d, ... nth term: aₙ = a₁ + (n−1)×d. Sum of first n terms: Sₙ = n/2 × (2a₁ + (n−1)d) = n/2 × (first + last). Example: 5, 8, 11, 14, 17 has d=3. The 10th term = 5 + 9×3 = 32. Sum of 10 terms = 10/2 × (5+32) = 185.

A geometric sequence multiplies each term by a fixed common ratio r: a₁, a₁r, a₁r², a₁r³, ... nth term: aₙ = a₁ × r^(n−1). Sum of first n terms: Sₙ = a₁ × (1−rⁿ)/(1−r) for r≠1. Example: 3, 6, 12, 24, 48 has r=2. The 8th term = 3 × 2^7 = 384. Sum of 8 terms = 3 × (1−256)/(1−2) = 765.

Fibonacci: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... Each term = sum of previous two: F(n) = F(n−1) + F(n−2). F(10) = 55. F(20) = 6765. F(50) = 12,586,269,025. The ratio of consecutive terms approaches the golden ratio φ ≈ 1.61803 as n grows. Enter any n in the Fibonacci tab.

Enter a comma-separated sequence (e.g. 2,4,8,16,32) and press Find Pattern. The calculator checks if the differences between terms are constant (arithmetic) or if the ratios are constant (geometric). If detected, it shows the formula, next term, and how to find any nth term. If neither pattern fits (e.g. prime numbers or polynomial sequences), it reports the pattern is not detected.

Sₙ = n/2 × (first + last) = 100/2 × (1+100) = 5050. Use Arithmetic mode: first term=1, common difference=1, n=100. This is the famous result attributed to Gauss who reportedly solved it instantly as a child by pairing: 1+100=101, 2+99=101, ..., 50+51=101, giving 50 pairs of 101 = 5050.

When |r| < 1: S∞ = a₁ / (1−r). Example: 1 + 1/2 + 1/4 + 1/8 + ... = 1/(1−0.5) = 2. Example: 1 + 1/3 + 1/9 + ... = 1/(1−1/3) = 1.5. For |r| ≥ 1 the series diverges — terms grow without bound. The calculator computes partial sums for any finite n; use the formula for the infinite case.

Arithmetic: check if the differences between consecutive terms are equal. Geometric: check if the ratios between consecutive terms are equal. Example [2,5,8,11]: differences are 3,3,3 → arithmetic with d=3. Example [2,6,18,54]: ratios are 3,3,3 → geometric with r=3. Use Find Pattern mode to automate this check.

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