Arithmetic Sequence
The nth term and sum of an arithmetic sequence.
10th term (aₙ)
29
sum of first 10 = 155
Sum of terms
155
Common difference
3
AI Breakdown & Smart Takeaway
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How the Arithmetic Sequence works
This calculator finds the nth term and the sum of any arithmetic sequence given a first term, common difference, and number of terms — making it essential for students, teachers, and anyone working with evenly spaced numerical patterns.
An arithmetic sequence is a list of numbers in which each term increases or decreases by a fixed amount called the common difference. For example, 3, 7, 11, 15 is an arithmetic sequence with a first term of 3 and a common difference of 4. This calculator takes those two values plus the number of terms you want to examine, then instantly computes any specific term in the sequence and the running total of all terms up to that point.
To find the nth term, the calculator applies the formula a(n) = a₁ + (n − 1)d, where a₁ is the first term, d is the common difference, and n is the position of the term you want. This formula works because each step forward in the sequence adds exactly one more common difference to the first term. A common mistake is to multiply n by d instead of (n − 1), which overcounts by one interval — for instance, the 1st term of any sequence must equal a₁, not a₁ + d.
The sum of the first n terms, often called an arithmetic series, is computed using S(n) = n/2 × (2a₁ + (n − 1)d), or equivalently S(n) = n/2 × (a₁ + aₙ). Both forms are mathematically identical; the second is more convenient when you already know the last term. This result traces back to a famous shortcut attributed to Gauss: pairing the first and last terms of a sequence always yields the same sum, and there are n/2 such pairs, so you simply multiply.
One subtlety worth understanding is that the common difference d can be negative, producing a decreasing sequence such as 100, 95, 90 … In this case the sum will eventually peak and then decline as you add more terms. The calculator handles negative differences and non-integer values seamlessly, which is useful in real-world applications like depreciation schedules, salary step-increments, stadium seating layouts, or any scenario where quantities change by a constant amount over discrete steps.
Formula
aₙ = a₁ + (n−1)d · Sₙ = n/2 · (2a₁ + (n−1)d)
Pro tips
- If you know two terms but not the common difference, calculate d = (aₖ − a₁) / (k − 1), where k is the position of the known later term, then enter a₁ and d into the calculator.
- To find how many terms you need for the sum to reach a target value S, rearrange S(n) = n/2 × (2a₁ + (n − 1)d) into a quadratic in n and solve — the calculator's sum output lets you quickly verify your answer by adjusting n.
- Watch the sign of d carefully: entering a positive number when you mean a decreasing sequence is the single most common input error and will produce a completely wrong nth term and sum.
- For large n, use the S(n) = n/2 × (a₁ + aₙ) form mentally as a sanity check — average the first and last term, then multiply by n. If the calculator's output doesn't match this estimate, recheck your inputs.
- Arithmetic sequences appear in loan amortization, uniform acceleration problems, and musical tuning scales — recognizing the constant-difference pattern in context helps you set up a₁ and d correctly before you even open the calculator.
Key terms
- First Term (a₁)
- — The starting value of the arithmetic sequence, from which all subsequent terms are derived by repeatedly adding the common difference.
- Common Difference (d)
- — The fixed amount added to each term to produce the next; it can be positive (increasing), negative (decreasing), or zero (constant sequence).
- nth Term
- — The value of the sequence at position n, calculated without listing every preceding term using the formula a(n) = a₁ + (n − 1)d.
- Arithmetic Series
- — The sum of a specified number of consecutive terms of an arithmetic sequence, as opposed to the sequence itself which is just the ordered list.
- Number of Terms (n)
- — The count of elements in the portion of the sequence being analyzed, which directly controls both which term is found and how many are summed.
- Partial Sum S(n)
- — The total obtained by adding the first n terms of an arithmetic sequence, computed efficiently using the Gauss summation formula rather than term-by-term addition.
