Geometric Sequence Calculator
Calculate nth term, finite and infinite sums, common ratio, and visualize exponential growth or decay in geometric sequences.
Arithmetic Sequence Calculator
Calculate the nth term, sum of n terms, common difference, and generate arithmetic sequences with visual growth charts and partial sums.
nth Term (aₙ)⧉
29.00
a₁ + (n−1)·d = 2 + (10−1)·3
Sum of n Terms (Sₙ)⧉
155.00
n/2 · (2a₁ + (n−1)d)
Common Difference⧉
3.00
Constant difference between consecutive terms
Number of Terms⧉
10
Total terms in the sequence
Average (Mean)⧉
15.50
Sum divided by number of terms
Last Term⧉
29.00
Final term in the generated sequence
Sequence Growth
a1
2.00
a2
5.00
a3
8.00
a4
11.00
a5
14.00
a6
17.00
a7
20.00
a8
23.00
a9
26.00
a10
29.00
Sequence & Partial Sums Table
| n | aₙ | Sₙ | aₙ / Sₙ |
|---|---|---|---|
| 1 | 2.00 | 2.00 | 1.0000 |
| 2 | 5.00 | 7.00 | 0.7143 |
| 3 | 8.00 | 15.00 | 0.5333 |
| 4 | 11.00 | 26.00 | 0.4231 |
| 5 | 14.00 | 40.00 | 0.3500 |
| 6 | 17.00 | 57.00 | 0.2982 |
| 7 | 20.00 | 77.00 | 0.2597 |
| 8 | 23.00 | 100.00 | 0.2300 |
| 9 | 26.00 | 126.00 | 0.2063 |
| 10 | 29.00 | 155.00 | 0.1871 |
Planning notes, formulas, and examples
About the Arithmetic Sequence Calculator
An arithmetic sequence (or arithmetic progression) is one of the most fundamental patterns in mathematics — a sequence of numbers where each term increases or decreases by a constant value called the common difference. From simple counting (1, 2, 3, …) to practical scenarios like monthly savings growth or evenly spaced measurements, arithmetic sequences appear throughout daily life and advanced math alike.
This calculator lets you find the nth term, compute the sum of any number of terms, determine the common difference between terms, or figure out how many terms exist between two values. It generates the full sequence, shows partial sums, and plots a growth chart so you can visualize how arithmetic progressions behave.
The general formula for the nth term is aₙ = a₁ + (n − 1)d, where a₁ is the first term and d is the common difference. The sum of the first n terms is Sₙ = n/2 · (2a₁ + (n − 1)d), also written as Sₙ = n/2 · (a₁ + aₙ). This sum formula was famously discovered by young Gauss when asked to add the integers from 1 to 100. These formulas are essential in algebra, number theory, financial mathematics, and many applied sciences.
When This Page Helps
Arithmetic sequence calculations involve multiple interrelated variables — first term, common difference, number of terms, nth term, and partial sums — and mistakes in one step cascade through the rest. This calculator solves for any unknown given the other parameters, generates full sequences with partial sums and growth charts, and shows the mean. Students use it to verify homework and build intuition for linear growth patterns, while teachers can demonstrate how changing the common difference reshapes the entire sequence in real time.
How to Use the Inputs
- Choose a calculation mode — find nth term, sum, common difference, or number of terms.
- Enter the first term (a₁) of your arithmetic sequence.
- Provide the common difference (d) or the last term, depending on the mode selected.
- Enter the number of terms (n) you want to analyze.
- Use preset buttons for common sequences like natural numbers or even numbers.
- Review the output cards for computed values and the growth chart for a visual representation.
- Scroll down to the table for a term-by-term breakdown with partial sums.
Formula used
aₙ = a₁ + (n − 1) · d
Sₙ = n/2 · (2a₁ + (n − 1) · d)
d = (aₙ − a₁) / (n − 1)Example Calculation
Result: a₁₀ = 29, S₁₀ = 155
Starting at 2 and adding 3 each time: 2, 5, 8, 11, …, 29. The 10th term is 2 + 9·3 = 29. The sum is 10/2 · (2 + 29) = 155.
Tips & Best Practices
- A negative common difference produces a decreasing sequence.
- The average of an arithmetic sequence always equals the average of its first and last terms.
- The sum formula is equivalent to (number of terms) × (average of first and last term).
- Arithmetic sequences model linear growth — constant additions per step.
- If you know any three of a₁, d, n, aₙ, you can find the fourth.
The Gauss Story and the Sum Formula
According to legend, young Carl Friedrich Gauss was asked to add the integers from 1 to 100 as busywork. He quickly answered 5,050 by pairing the first and last terms (1+100, 2+99, …), forming 50 pairs of 101. This insight generalizes to the arithmetic series sum formula Sₙ = n/2 · (a₁ + aₙ). The formula works because the average term in any arithmetic sequence is exactly the mean of the first and last terms, making the total equal to that average multiplied by the count. This elegant relationship is the basis for many results in combinatorics and number theory.
Arithmetic Sequences in Real Life
Arithmetic sequences model any situation with constant change per step. Salary raises of a fixed dollar amount each year, evenly spaced seats in an auditorium (each row has two more seats than the one in front), linear depreciation of equipment value, uniform acceleration in physics (velocity increases by a constant each second), and numbering systems where items are labeled at regular intervals all follow arithmetic progressions. Recognizing these patterns lets you apply closed-form formulas instead of adding up terms one by one.
Common Pitfalls and How to Avoid Them
The most common mistake is confusing (n − 1) with n in the nth-term formula: aₙ = a₁ + (n − 1)d, not a₁ + n·d. Another frequent error is using the wrong formula for the sum — remember that Sₙ = n/2 · (2a₁ + (n−1)d) includes the factor of 2 in front of a₁. When finding the common difference from two known terms, always divide the gap by (n − 1), not n. Finally, double-check signs: a negative d means the sequence decreases, and the sum can become negative if the sequence crosses zero.
Sources & Methodology
Last updated:
Frequently Asked Questions
A sequence where each term differs from the previous one by a constant called the common difference (d). Example: 3, 7, 11, 15, … has d = 4.
Subtract any term from the next: d = a(n+1) − aₙ. Alternatively, d = (aₙ − a₁) / (n − 1) if you know the first and last terms.
Sₙ = n/2 · (2a₁ + (n − 1)d), or equivalently Sₙ = n · (a₁ + aₙ) / 2.
Yes — a constant sequence like 5, 5, 5, 5, … is a valid arithmetic sequence with d = 0.
Arithmetic sequences add a constant (linear growth); geometric sequences multiply by a constant (exponential growth).
Budgeting with regular deposits, evenly spaced time intervals, linear depreciation, seating arrangements in auditoriums, and many physics problems involving constant acceleration.
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