Find the nth term and the sum of an arithmetic progression. Calculate sequence values for any arithmetic series.
nth Term (aₙ)
32
Sum of n Terms (Sₙ)
185
Enter valid numeric values for a₁, d, and positive integer n
An arithmetic sequence (or arithmetic progression) is a sequence of numbers in which the difference between consecutive terms is always the same. This constant difference is called the "common difference" and is denoted by d.
For example, the sequence 2, 4, 6, 8, 10 is arithmetic with a₁ = 2 and d = 2. Each term is obtained by adding d to the previous term. Arithmetic sequences are fundamental in mathematics and appear in many real-world contexts, from calculating loan payments to predicting population growth.
The common difference can be positive (increasing sequence), negative (decreasing sequence), or zero (constant sequence). Arithmetic sequences have beautiful mathematical properties that allow us to efficiently calculate any term or the sum of multiple terms without computing all intermediate values.
Finds any term in the sequence directly without calculating previous terms.
Calculates the sum of the first n terms efficiently.
Find the 10th term and sum of the sequence: 5, 8, 11, 14...
Step 1: Identify the known values
a₁ = 5 (first term)
d = 3 (common difference: 8 - 5 = 3)
n = 10 (we want the 10th term)
Step 2: Calculate the 10th term using aₙ = a₁ + (n - 1)d
a₁₀ = 5 + (10 - 1) × 3
a₁₀ = 5 + 27 = 32
Step 3: Calculate the sum using Sₙ = (n/2)(a₁ + aₙ)
S₁₀ = (10/2)(5 + 32)
S₁₀ = 5 × 37 = 185
Answer: The 10th term is 32, and the sum of the first 10 terms is 185.
Yes. If d is negative, the sequence decreases. For example, 10, 7, 4, 1... has d = -3.
A sequence is a list of ordered numbers. A series is the sum of those numbers.
Use d = (aₘ - aₙ) / (m - n), where aₘ and aₙ are the m-th and n-th terms respectively.
Yes: Sₙ = (n/2)[2a₁ + (n-1)d]. Both versions give identical results.
All terms are the same (constant sequence). The nth term equals a₁, and the sum is n × a₁.
Use the nth term formula aₙ = a₁ + (n - 1)d directly. This is one major advantage of arithmetic sequences.
Typically, n is a positive integer representing term position. Fractions or negatives don't have standard interpretations.
Loan payments, savings plans, depreciation schedules, temperature changes, and any linear growth/decay pattern.
Related Tools