Partial Sum of Arithmetic Series
🧮 Partial Sum Calculator – Quickly Find the Sum of the First n Terms of a Sequence
A Partial Sum Calculator helps you compute the sum of the first n terms of a sequence or series. This is useful in both mathematics and statistics, especially when dealing with arithmetic or geometric sequences, or preparing for exams involving series.
📘 What Is a Partial Sum?
A partial sum is the sum of a specified number of terms of a sequence, usually starting from the first term.
For a sequence a1,a2,a3,…,ana_1, a_2, a_3, …, a_na1,a2,a3,…,an, the n-th partial sum is: Sn=a1+a2+a3+⋯+anS_n = a_1 + a_2 + a_3 + \cdots + a_nSn=a1+a2+a3+⋯+an
🔢 Types of Sequences Supported
| Type | Formula for Partial Sum |
|---|---|
| Arithmetic Series | Sn=n2(2a+(n−1)d)S_n = \frac{n}{2}(2a + (n – 1)d)Sn=2n(2a+(n−1)d) |
| Geometric Series | Sn=a⋅1−rn1−rS_n = a \cdot \frac{1 – r^n}{1 – r}Sn=a⋅1−r1−rn (if r≠1r \ne 1r=1) |
| Custom Series | Enter formula for general term ana_nan |
✅ Example: Arithmetic Partial Sum
Find the sum of the first 5 terms of the sequence:
3, 6, 9, 12, 15 (common difference d=3d = 3d=3) S5=52(2×3+(5−1)×3)=52(6+12)=52×18=45S_5 = \frac{5}{2} (2×3 + (5-1)×3) = \frac{5}{2}(6 + 12) = \frac{5}{2} × 18 = 45S5=25(2×3+(5−1)×3)=25(6+12)=25×18=45
✅ Example: Geometric Partial Sum
Find the sum of the first 4 terms of:
2, 4, 8, 16 (common ratio r=2r = 2r=2) S4=2×1−241−2=2×1−16−1=2×(−15)/(−1)=30S_4 = 2 × \frac{1 – 2^4}{1 – 2} = 2 × \frac{1 – 16}{-1} = 2 × (-15)/(-1) = 30S4=2×1−21−24=2×−11−16=2×(−15)/(−1)=30
🧰 Features of a Good Partial Sum Calculator
- Supports arithmetic, geometric, and formula-based sequences
- Accepts general term input (e.g., an=n2+1a_n = n^2 + 1an=n2+1)
- Allows you to specify:
- First term or general rule
- Number of terms (
n) - Common difference or ratio
- Displays:
- Step-by-step breakdown
- Exact sum and simplified form
- Optionally: visual or tabular representation
❓ FAQs – Partial Sum Calculator
🔹 What is the difference between partial sum and full sum?
A partial sum only includes the first n terms. A full sum refers to either the entire finite sequence or the infinite series sum, if it converges.
🔹 Can I use custom formulas like an=n2a_n = n^2an=n2?
Yes — most calculators allow you to input a formula for the nnn-th term and will compute the sum from 1 to n.
🔹 Does the calculator support infinite series?
Partial sums apply to both finite and converging infinite series — as you increase n, the partial sum approaches the total.
🔹 Is the calculator useful for statistics?
Yes. In descriptive statistics, partial sums help compute cumulative totals and moving averages.
🔹 How many terms should I enter?
You choose n based on how far you want to sum. The more terms, the closer you get to the total (in infinite series).