Math & Algebra

Sequence & Series Calculator — Arithmetic (AP) & Geometric (GP) Progressions

Mathematical sequences and series underpin discrete mathematics, calculus, financial compound interest, and algorithmic recurrence relations. Our free Sequence & Series Calculator solves both Arithmetic Progressions (AP) and Geometric Progressions (GP). Compute the n-th term (a_n), sum of the first n terms (S_n), common difference (d), common ratio (r), and the sum of an infinite geometric series with interactive term tables.

Sequence & Series Calculator

Arithmetic (AP), Geometric (GP), Harmonic (HP) & Fibonacci Sequences

First Term (a₁)

Common Difference (d)

Term Number (n)

Sequence Results (AP)

10-th Term (a₍10₎)

Sum of First 10 Terms (S₍10₎)

155

Average of Terms

15.5000

Step-by-Step Mathematical Substitution:

aₙ = a₁ + (n - 1)d ➔ a₍10₎ = 2 + (10 - 1)(3) = 29

First 10 Terms of the Sequence

Showing terms 1 to 10

Term 1

Term 2+3

Term 3+3

Term 4+3

Term 5+3

Term 6+3

Term 7+3

Term 8+3

Term 9+3

Term 10+3

Mathematical Formulas & Calculation Methods

Arithmetic Progression (AP) Formulas

n-th term and sum of first n terms for constant difference d.

a_n = a₁ + (n-1)d | S_n = (n/2)(2a₁ + (n-1)d)

Geometric Progression (GP) Formulas

n-th term and sum of first n terms for constant multiplier r.

a_n = a₁ × r^(n-1) | S_n = a₁(1 - r^n) / (1 - r) (r ≠ 1)

Infinite Geometric Series Sum

Sum of infinite GP terms when |r| < 1.

S_∞ = a₁ / (1 - r) provided |r| < 1

Worked Step-by-Step Examples

Example 1: Arithmetic Progression 15th Term & Sum

"For an AP with first term a₁=5 and common difference d=4, calculate the 15th term (a₁₅) and sum of first 15 terms (S₁₅)."

Steps:

Calculate 15th term: a₁₅ = 5 + (15 - 1) × 4 = 5 + (14 × 4) = 5 + 56 = 61.
Calculate sum S₁₅: S₁₅ = (15 / 2) × (5 + 61) = 7.5 × 66 = 495.

Result: 15th Term a₁₅ = 61, Sum S₁₅ = 495

Example 2: Infinite Geometric Series Sum

"Calculate the infinite series sum for a GP with a₁=12 and common ratio r=0.5."

Steps:

Check convergence condition: |r| = 0.5 < 1, so the infinite series converges.
Apply formula: S_∞ = a₁ / (1 - r) = 12 / (1 - 0.5) = 12 / 0.5 = 24.

Result: Infinite Series Sum S_∞ = 24

What is Sequence & Series Calculator (AP & GP)?

A sequence calculator is an online algebra solver that computes terms and series sums for numerical progressions. A sequence is an ordered list of numbers following a specific mathematical rule. A series is the sum of the terms of a sequence.

In an Arithmetic Progression (AP), the difference between consecutive terms remains constant; this difference is called the common difference (d). In a Geometric Progression (GP), each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r).

When evaluating infinite geometric series, the series is said to converge to a finite sum if the absolute value of the common ratio is strictly less than 1 (|r| < 1). If |r| ≥ 1, the series diverges to infinity.

How to Use Sequence & Series Calculator (AP & GP)

1Select your progression type: Arithmetic Progression (AP) or Geometric Progression (GP).
2Input the first term (a₁), common difference (d) or common ratio (r), and number of terms (n).
3Click Calculate to view the n-th term, partial sum S_n, and infinite sum S_∞ (for GP).
4Explore the generated term preview table showing individual sequence terms.

Key Features

Dual mode solver for Arithmetic Progressions (AP) and Geometric Progressions (GP).
n-th term finder (a_n) and sum of first n terms solver (S_n).
Infinite geometric series convergence testing and sum calculator.
Interactive sequence term preview table generator.
Step-by-step arithmetic breakdown.

Benefits of Using Sequence & Series Calculator (AP & GP)

Eliminates tedious manual term-by-term addition for long sequences.

Clarifies AP vs GP formulas for algebra and calculus students.

Calculates compound interest sequence series effortlessly.

Helps analyze convergence vs divergence of infinite series.

Practical Use Cases

Algebra & Precalculus Students

Solve homework problems on AP/GP formulas, sequence patterns, and series sums.

Finance Students & Investors

Model annuity payments, compound interest series, and loan amortization sequences.

Computer Science Students

Analyze algorithm loop counts, memory expansion series, and recursive functions.

Physics Students

Calculate bouncing ball height decay series and harmonic oscillator steps.

Frequently Asked Questions

What is the main difference between an AP and a GP?

In an Arithmetic Progression (AP), consecutive terms differ by adding a fixed constant (d). In a Geometric Progression (GP), consecutive terms differ by multiplying by a fixed ratio (r).

How do you find the common ratio (r) of a GP?

Divide any term in the GP by its preceding term: r = a_k / a_{k-1}. For example, in 3, 6, 12, 24, the common ratio r is 6/3 = 2.

When does an infinite geometric series converge?

An infinite geometric series converges to a finite sum S_∞ = a₁ / (1 - r) if and only if the absolute value of the common ratio is strictly less than 1 (|r| < 1).

What is the n-th term formula for an Arithmetic Progression?

The n-th term formula for an AP is a_n = a₁ + (n - 1)d, where a₁ is the first term and d is the common difference.

What happens to a GP when r = 1?

When r = 1, every term in the GP is identical to the first term (a₁). The sum of n terms is simply S_n = n × a₁.

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Math & Algebra

Sequence & Series Calculator — Arithmetic (AP) & Geometric (GP) Progressions