Arithmetic & Geometric Sequence Calculator – AP & GP

Our Arithmetic and Geometric Sequence Calculator helps students, teachers, and math enthusiasts calculate the nth term, sum of terms, common difference (AP), and common ratio (GP) quickly and accurately.

What is an Arithmetic Sequence (AP)?

An Arithmetic Sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference (d).

Arithmetic Sequence Formulas

• nth term (Tn) = a + (n-1)d
• Sum of n terms (Sn) = n/2 × [2a + (n-1)d]
• a = first term
• d = common difference
• n = number of terms

Example 1 – Arithmetic Sequence

AP: 2, 5, 8, 11 … | Find 10th term & sum of first 10 terms

a = 2, d = 3, n = 10
T10 = 2 + (10-1)*3 = 2 + 27 = 29
S10 = 10/2 × [2*2 + (10-1)*3] = 5 × [4 + 27] = 5 × 31 = 155

Tips for Arithmetic Sequence

• Identify first term (a) and common difference (d) clearly.
• Use formulas to avoid manual addition for large n.
• Check consistency of sequence before applying formula.

What is a Geometric Sequence (GP)?

A Geometric Sequence is a sequence where each term is obtained by multiplying the previous term by a constant called common ratio (r).

Geometric Sequence Formulas

• nth term (Tn) = a × r^(n-1)
• Sum of n terms (Sn) = a × (r^n – 1) / (r – 1), r ≠ 1
• Sum of infinite terms (S∞) = a / (1 – r), |r| < 1

Example 2 – Geometric Sequence

GP: 3, 6, 12, 24 … | Find 8th term & sum of first 8 terms

a = 3, r = 2, n = 8
T8 = 3 × 2^(8-1) = 3 × 128 = 384
S8 = 3 × (2^8 – 1) / (2 – 1) = 3 × (256-1)/1 = 3 × 255 = 765

Tips for Geometric Sequence

• Identify first term (a) and common ratio (r) correctly.
• For decreasing GP, |r| < 1, infinite sum can be calculated.
• Be careful with negative or fractional common ratios.

Arithmetic & Geometric Sequence Calculator Features

• Calculate nth term of AP or GP instantly.
• Calculate sum of n terms or infinite terms.
• Supports positive, negative, and fractional terms.
• Ideal for students, teachers, engineers, and finance calculations.
• Free, online, and mobile-friendly tool.

Worked Example – Mixed AP & GP

AP: 4, 7, 10, … | GP: 2, 6, 18, …

AP – T20 = 4 + (20-1)*3 = 4 + 57 = 61
AP – S20 = 20/2 × [2*4 + 57] = 10 × 65 = 650
GP – T5 = 2 × 3^(5-1) = 2 × 81 = 162
GP – S5 = 2 × (3^5 – 1)/(3-1) = 2 × (243-1)/2 = 242

FAQs – Arithmetic & Geometric Sequence Calculator

1. Can I use negative numbers in sequences?

Yes, both AP and GP can have negative numbers.

2. Can the common ratio be fractional?

Yes, GP can have fractional or decimal ratios.

3. How to find sum of infinite geometric sequence?

Use S∞ = a / (1 – r) where |r| < 1.

4. What is the difference between AP and GP?

AP has constant difference; GP has constant ratio.

5. Can this calculator help in finance calculations?

Yes, geometric sequences are often used in finance for compound interest calculations.

Advantages of Using Sequence Calculator

• Quickly calculates nth term and sum of terms.
• Reduces manual calculation errors.
• Supports both arithmetic and geometric sequences.
• Easy for homework, exams, and professional calculations.
• Free and accessible online anywhere.

Final Thoughts

The Arithmetic & Geometric Sequence Calculator is an essential tool for students, teachers, and math enthusiasts. It simplifies calculations, ensures accuracy, and saves time for both academic and practical applications.

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