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Arithmetic Sequence Calculator

Find the nth term and the sum of the first n terms of an arithmetic sequence from its first term and common difference.

Nth Term aₙ = a₁ + (n - 1)d
48
First terms: 3, 8, 13, 18, 23, 28, 33, 38, 43, 48 ...
Sum of First n Terms (Sₙ)
255
Formula: Sₙ = (n/2)(2a₁ + (n-1)d)
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Calculation Guide & Reference

Arithmetic Sequence Mathematics

Arithmetic sequence calculator finds the nth term and the sum of the first n terms from the first term and common difference.

Standardized Mathematical Formula
aₙ = a₁ + (n − 1)d

In an arithmetic sequence, each term increases by a fixed common difference d. The nth term is the first term plus (n−1) increments of d. The sum of the first n terms is n/2 times the sum of the first and last term.

Variables:
a₁:First term
d:Common difference between consecutive terms
n:Term number
How It Works (Step-by-Step)
  • 1Enter the first term, common difference, and the term number you want.
  • 2View the nth term value and the sum of the first n terms.
Real-World Numerical Example
First Term 3, Difference 5, 10th Term

An arithmetic sequence starts at 3 with a common difference of 5. Find the 10th term and the sum of the first 10 terms.

•10th term: 3 + (10 − 1) × 5 = 3 + 45 = 48.
•Sum: (10/2) × (2 × 3 + (10 − 1) × 5) = 5 × (6 + 45) = 5 × 51 = 255.
Result: The 10th term is 48, and the sum of the first 10 terms is 255.
Calculation Best Practices & Tips
Examine the discriminant (b² - 4ac) before solving quadratics to instantly know if roots are real, distinct, repeated, or complex.
In statistical analysis, always check both the mean and median to identify if data distribution is skewed by outliers.
When calculating percentage changes, ensure you divide by the original starting baseline value: ((New - Old) / Old) * 100.

Frequently Asked Questions (FAQ)

A sequence where each term differs from the previous one by a constant amount, called the common difference (d). Example: 3, 8, 13, 18... has d = 5.

Sₙ = (n/2)(2a₁ + (n−1)d), or equivalently (n/2)(a₁ + aₙ) if you already know the last term.

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