Sum of a Linear Number Sequence Calculator
Find the sum of any arithmetic sequence using the first term, common difference, and number of terms.
Enter the first term, common difference, and number of terms to compute the sum of the arithmetic sequence instantly.
Sum of a Linear Number Sequence Calculator
Find the sum of any arithmetic sequence using the first term, common difference, and number of terms.
About the Sum of a Linear Number Sequence Calculator
A linear number sequence — also called an arithmetic sequence or arithmetic progression — is a sequence of numbers where each term after the first is obtained by adding a fixed constant called the common difference to the previous term. If the first term is a and the common difference is d, the sequence runs: a, a+d, a+2d, a+3d, and so on up to the nth term.
The general term (the nth term) of an arithmetic sequence is given by the formula: an = a + (n − 1)d. This tells you the value of any term without computing all preceding terms. For example, in the sequence 2, 5, 8, 11, 14 (first term 2, common difference 3), the 10th term is 2 + (10 − 1) × 3 = 29.
The sum of the first n terms, denoted Sn, is calculated using the formula: Sn = n/2 × [2a + (n − 1)d]. This elegant formula can also be written as Sn = n/2 × (first term + last term), which is particularly useful when the last term is known. The formula was famously used by Gauss as a child to quickly add the integers from 1 to 100: n=100, a=1, d=1, so S100 = 100/2 × (1 + 100) = 5050.
Arithmetic sequences have a constant rate of change, which means when you plot their terms on a graph, they form a straight line — hence the name linear sequence. This is in contrast to geometric sequences, where terms have a constant ratio and the graph curves exponentially.
In the real world, arithmetic sequences model many practical situations. A salary that increases by a fixed amount each year forms an arithmetic sequence. The distance traveled in each successive second by an object accelerating at a constant rate is arithmetic. Stacking rows of seats in a theater, where each row has one more seat than the row in front, creates an arithmetic sequence. Annuity calculations, simple interest computations, and linear depreciation all rely on the same arithmetic progression formulas this calculator uses.
Arithmetic Sequence Sum Examples
Common examples illustrating the sum formula for arithmetic sequences.
| Input (a, d, n) | Sum (Sn) | Notes |
|---|---|---|
| a=1, d=1, n=100 | 5050 | Sum of integers 1 to 100. Sn = 100/2 × (1+100) = 50 × 101 = 5050. The classic Gauss problem. |
| a=2, d=3, n=5 | 40 | Sequence: 2, 5, 8, 11, 14. Sn = 5/2 × [2×2 + (5−1)×3] = 2.5 × 16 = 40. |
| a=10, d=−3, n=4 | 22 | Decreasing sequence: 10, 7, 4, 1. Sn = 4/2 × [20 + 3×(−3)] = 2 × 11 = 22. |
| a=5, d=0, n=6 | 30 | Constant sequence: d=0 means all terms equal 5. Sum = 6 × 5 = 30. |
How to Use the Arithmetic Sequence Sum Calculator
- Enter the first term (a) — the value of the first number in your sequence.
- Enter the common difference (d) — the fixed amount added to each term. Use a negative value for a decreasing sequence.
- Enter the number of terms (n) — how many terms in the sequence you want to sum. Must be a positive integer.
- Click 'Calculate Sum'. The calculator displays the sum Sn, the last term an, and the formula used.
- Click 'Reset' to clear all fields and start a new calculation.
Arithmetic Sequence Sum FAQ
What is the formula for the sum of an arithmetic sequence?
The formula is Sn = n/2 × [2a + (n − 1)d], where n is the number of terms, a is the first term, and d is the common difference. Equivalently, Sn = n/2 × (first term + last term). Both forms give the same result — use whichever is more convenient given the information you have.
Can the common difference be negative or zero?
Yes. A negative common difference means the sequence is decreasing — each term is smaller than the previous one. For example, 10, 7, 4, 1 has d = −3. A common difference of zero means all terms are identical and the sum equals n × a.
What is the difference between an arithmetic sequence and a geometric sequence?
In an arithmetic sequence, terms differ by a constant addition (common difference d). In a geometric sequence, terms differ by a constant multiplication (common ratio r). Arithmetic sequences grow linearly; geometric sequences grow exponentially. This calculator is designed specifically for arithmetic (linear) sequences.
How do I find the number of terms when I know the first term, last term, and common difference?
Use the formula n = (last term − first term) / d + 1. For example, in the sequence 3, 7, 11, 15, 19, the last term is 19, first term is 3, and d is 4: n = (19 − 3) / 4 + 1 = 5. Once you know n, enter a, d, and n into the calculator to find the sum.
Why does the sum formula use n/2?
The n/2 factor comes from pairing the first and last terms, which always sum to the same value. When you write the sequence forward and backward and add corresponding terms, each pair equals (first term + last term). There are n such pairs split between the two copies, so you multiply by n/2.
Can this calculator be used for simple interest calculations?
Yes. Simple interest on a loan or investment generates an arithmetic sequence of balances. If you start with principal P, earn interest I each period, and want the total after n periods, set a = P + I, d = I, and n to the number of periods. The sum gives the total of all period-end balances.