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Arithmetic Series Sum
Calculates the sum of the first n terms of an arithmetic sequence from the first term, last term, and number of terms.
Interpretation
S_n = (n/2)(a₁ + aₙ). Sum of an arithmetic sequence. a₁ first term, aₙ nth term. Used in problems with constant increments.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| S_n | Sum of the first n terms | |
| n | Number of terms | |
| a1 | First term of the sequence | |
| an | Last (n-th) term of the sequence |
What it means
The sum of the first n terms of an arithmetic sequence (where the difference between consecutive terms is constant) is given by S_n = n/2 (a₁ + aₙ), where a₁ is the first term and aₙ is the nth term. This formula is derived by pairing terms from the beginning and end. It is used in many real‑world problems: calculating total distance travelled with constant acceleration, total cost with linear pricing, and summing sequences in computer algorithms (e.g., for loop iterations). Understanding the arithmetic series sum is basic to algebra and has applications in finance (linear depreciation) and engineering (linear interpolation).
Worked example
Arithmetic Series Sum – Two Examples
Real‑World| Parameter | Value |
|---|---|
| n | 10 |
| a₁ | 1 |
| aₙ | 10 |
| Parameter | Value |
|---|---|
| n | 20 |
| a₁ | 5 |
| aₙ | 24 |
Common mistakes
- Arithmetic series sum: Sₙ = (n/2)(a₁ + aₙ) – sum of n terms.
- Number of terms: n must be an integer.
- First and last: a₁ and aₙ are the first and last terms – if only the common difference is given, compute aₙ = a₁ + (n−1)d.
- Average: The sum is n times the average of the first and last terms.
- Check: For small n, verify manually.
Applications
The sum of an arithmetic series, S_n = (n/2)(a₁ + aₙ), gives the total of a sequence with constant difference. This is used in scheduling, resource allocation, and cost estimation. Engineers use it to calculate total distance travelled with constant acceleration, to sum incremental costs, and to design step‑wise processes. In computer science, it appears in algorithm analysis for loop sums. In business, it calculates total sales over periods with linear growth. By using this formula, professionals can quickly sum large sequences without manual addition, enabling efficient planning and optimisation.
- Accumulation of linear sequences (e.g., total distance in constant acceleration)
- Scheduling of tasks with uniform increments
- Cost estimation for stepped production or procurement
- Algorithm analysis (sum of loops and complexity)
- Business – cumulative sales or production over time
Frequently Asked Questions
The sum of the first n terms of an arithmetic sequence with first term a₁ and last term aₙ is S_n = n/2 (a₁ + aₙ). Equivalently, S_n = n/2 (2a₁ + (n–1)d), where d is the common difference.
Applying the geometric series formula to a sequence that is actually arithmetic, since the two series types require different formulas.
A sequence where each term differs from the previous one by a constant d: a, a+d, a+2d, ... .
Using the formula: S = 100/2 (1 + 100) = 50 × 101 = 5050.
aₙ = a₁ + (n–1)d.
If you know the first term, last term, and common difference, you can find n from the nth term formula and then use the sum formula.
The average is (a₁ + aₙ)/2, which equals the sum divided by n.
- Calculating total earnings with constant increments.
- Summing costs that increase linearly.
- Approximating areas under a curve.
The formula still works; just use the given value of d.
The sum of the first n odd numbers is n².