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Sum of Linear Number Sequence Calculator

An arithmetic sequence—where each term increases or decreases by a constant amount—appears frequently in finance, scheduling, and data analysis. Rather than manually adding dozens or hundreds of terms, you can derive the exact sum using a single formula. This calculator accepts your starting value, the common difference, and how many terms exist, then instantly computes the total.

Last updated: April 29, 2026

Creators Anna Szczepanek, PhD Mathematics

Reviewers Mateusz Mucha and Jasmine J. Mah

2,480 people find this calculator helpful

The Arithmetic Sum Formula

The sum of a linear sequence depends on three quantities: the first term, the constant step between consecutive terms, and the total count of terms. Rather than adding each value individually, you can use an elegant closed-form expression.

S = (n ÷ 2) × (2a + d × (n − 1))

OR equivalently:

S = (n ÷ 2) × (a + l)

S — Sum of all terms in the sequence
n — Number of terms (periods)
a — First term of the sequence
d — Common difference between consecutive terms
l — Last (nth) term of the sequence

Understanding Linear Sequences

A linear sequence, also called an arithmetic progression, is any ordered list where the gap between adjacent numbers remains constant. Examples include:

2, 4, 6, 8, 10 (difference of 2)
100, 95, 90, 85 (difference of −5)
1.5, 2.0, 2.5, 3.0 (difference of 0.5)

The difference can be positive (increasing sequence), negative (decreasing sequence), or even fractional. Once you know the starting value, the difference, and how many terms exist, the entire sequence is determined. The beauty of the formula is that it avoids summing every single term—critical when working with hundreds or thousands of elements.

Step-by-Step Calculation

Method 1: Using first term and common difference

Identify your starting term a and common difference d.
Count the total number of terms n.
Calculate the last term: l = a + d × (n − 1).
Apply the formula: S = (n ÷ 2) × (a + l).

Method 2: Direct formula substitution

Plug a, d, and n into: S = (n ÷ 2) × (2a + d × (n − 1)).
Perform the arithmetic in the correct order.

Real-World Example: Subscription Revenue

Suppose a SaaS company offers tiered annual plans where each tier costs $100 more than the previous one. The entry-level plan costs $500, and they have 12 subscription tiers:

Tier 1: $500
Tier 2: $600
Tier 3: $700
⋮
Tier 12: $1,600

Using the formula with a = 500, d = 100, and n = 12:

S = (12 ÷ 2) × (500 + 1,600) = 6 × 2,100 = 12,600

The total revenue across all tiers is $12,600, without manually listing and adding each one.

Common Pitfalls and Practical Notes

When working with arithmetic sequences, watch for these frequent mistakes and considerations.

Counting terms correctly — Ensure you count both endpoints. The sequence 10, 12, 14, 16 contains 4 terms, not 3. A common error is counting only the "gaps" rather than the actual elements themselves.
Handling negative differences — Sequences can decrease (negative difference). The formula works identically—just ensure you apply the sign correctly when computing intermediate steps. For 50, 45, 40, 35, use d = −5, not 5.
Verifying your last term — Always double-check the final term using l = a + d × (n − 1) before entering it into the sum formula. A miscalculation here propagates to the final answer.
Non-integer differences — Arithmetic sequences need not involve whole numbers. Differences like 0.25 or 1/3 are perfectly valid. The formula remains unchanged; just preserve precision in your calculations.

Frequently Asked Questions

How is the sum formula for arithmetic sequences derived?

The derivation hinges on a clever pairing trick. If you write the sequence forwards and backwards, then add term-by-term, each pair sums to the same value: (first term + last term). With n pairs, the total is n × (first + last). Dividing by 2 (since we counted each term twice) yields S = (n ÷ 2) × (first + last). This elegant insight avoids brute-force addition entirely.

Can I use this calculator if I only know the first term and the last term?

Yes. If you know the first term a, the last term l, and the number of terms n, use the simplified form: S = (n ÷ 2) × (a + l). You don't need to enter the common difference at all. The calculator will compute it for you as d = (l − a) ÷ (n − 1) if needed for verification.

What is the sum of the integers from 1 to 100?

The natural numbers 1, 2, 3, …, 100 form an arithmetic sequence with a = 1, d = 1, and n = 100. Using the formula: S = (100 ÷ 2) × (1 + 100) = 50 × 101 = 5,050. This result is famous and often attributed to the young mathematician Carl Friedrich Gauss, who recognised the pattern as a child.

Why is the common difference sometimes called the "common ratio"?

It isn't—these are distinct concepts. A common difference (arithmetic sequence) means you add a fixed number each time. A common ratio (geometric sequence) means you multiply by a fixed number each time. For instance, 2, 4, 8, 16 has a common ratio of 2, not a common difference. This calculator handles only arithmetic sequences with constant differences.

Can the common difference be zero?

Yes, technically. If d = 0, every term equals the first term a. The sequence becomes a, a, a, … and the sum is simply S = n × a. While degenerate, the formula still applies: S = (n ÷ 2) × (2a + 0) = n × a.

How do fractional or decimal sequences affect the calculation?

They don't—the formula is indifferent to whether your values are integers, decimals, or fractions. A sequence like 0.5, 1.0, 1.5, 2.0, 2.5 with d = 0.5 works identically to whole-number sequences. Simply maintain adequate decimal precision throughout your computation, especially when the common difference is not a whole number or when dividing by 2.

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