Arithmetic Series
Simple Explanation
The sum of the first n terms of an arithmetic progression, Sₙ, can be found directly with Sₙ = n/2·(a₁+aₙ), without adding every term individually — the average of the first and last term, times how many terms there are.
Why Do We Need It?
This turns a potentially huge addition problem (like summing 1000 terms) into a single, quick calculation.
Formula
Sum of an Arithmetic Series
Sₙ = n/2 · (a₁ + aₙ) = n/2 · (2a₁ + (n − 1)d)
Gives the sum of the first n terms of an arithmetic sequence directly, without adding every term one at a time.
- Sₙ
- — the sum of the first n terms
- a₁, aₙ
- — the first and nth (last) terms being summed
- n
- — the number of terms
- d
- — the common difference
When to use it: Whenever you need the total of a run of arithmetic-sequence terms, without adding them individually.
Worked Example
Find the sum of an arithmetic series
Find the sum of the first 20 terms of the A.P. with a₁=3 and d=5.
Why Does This Work?
Pairing the first term with the last (a₁+aₙ), the second term with the second-to-last (a₂+aₙ₋₁), and so on, every pair adds up to exactly the same total, a₁+aₙ (since each step gains d going up while losing d going down) — with n/2 such pairs, the sum is n/2 times that constant pair-total.
Real-Life Example
Total attendance across a ticket price range
A theatre sells tickets at prices increasing by a fixed amount for each of 15 rows, and wants the total revenue.
If revenue per row forms an A.P., the arithmetic series formula gives the total revenue in one calculation, instead of adding 15 separate amounts.
Practice
Find the sum of the first 10 positive integers (1+2+...+10) using the arithmetic series formula.
MediumCommon mistake
Forgetting to find aₙ first before applying Sₙ=n/2(a₁+aₙ) — you need both the first AND last term of the range being summed.
Quick Review
- Sₙ = n/2 · (a₁ + aₙ) = n/2 · (2a₁ + (n−1)d).
- Pairing first-and-last terms gives n/2 pairs, each summing to the same total.
- Find aₙ first if it is not already given.