Arithmetic Progression (A.P.)
Simple Explanation
An arithmetic progression (A.P.) is a sequence where the same fixed number, the common difference d, is added to get from each term to the next. Its nth term is given directly by aₙ = a₁ + (n−1)d.
Why Do We Need It?
A.P.s model any process that changes by the same fixed amount every step — the most common and intuitive pattern in a sequence.
Formula
nth Term of an Arithmetic Progression
aₙ = a₁ + (n − 1)d
Gives the value of any term directly from its position n, without listing every earlier term.
- aₙ
- — the nth term of the sequence
- a₁
- — the first term
- d
- — the common difference — the fixed amount added between consecutive terms
- n
- — the term's position in the sequence
When to use it: Whenever you need a specific term of an arithmetic sequence directly, or need to find n, a₁, or d from other known values.
Worked Example
Find a term of an arithmetic progression
An A.P. has first term 5 and common difference 3. Find the 10th term.
Why Does This Work?
Starting at a₁ and adding d exactly (n−1) times reaches the nth term — since going from a₁ to a₂ is one addition, a₁ to a₃ is two additions, and so on, reaching aₙ always takes exactly n−1 additions of d.
Real-Life Example
Stadium seating rows
A stadium's rows each have 4 more seats than the row before, starting with 20 seats in row 1.
The number of seats per row forms an A.P. with a₁=20, d=4 — the nth term formula instantly gives the seat count for any row.
Practice
An A.P. has first term 7 and common difference −2. Find the 8th term.
MediumAn A.P. has a₁=4 and a₅=20. Find the common difference d.
MediumCommon mistake
Using n instead of (n−1) in the formula — the first term a₁ requires ZERO additions of d, not one, since (1−1)=0.
Quick Review
- aₙ = a₁ + (n−1)d.
- d = common difference: the fixed amount added between consecutive terms.
- Reaching the nth term always takes exactly (n−1) additions of d from a₁.