Arithmetic Progression: The Pattern of Equal Steps
Imagine you're climbing a staircase where every step has the exact same height. If the first step takes you to 3 feet, and each step after that adds exactly 2 feet, your heights would be: 3, 5, 7, 9, 11, ... That's an arithmetic progression — a sequence where you move forward by adding the same number every time.
The Core Idea
An Arithmetic Progression (AP) is a list of numbers where the difference between any two consecutive terms is constant. This constant is called the common difference, usually denoted by d.
If the first term is a, then the sequence looks like:
a, a+d, a+2d, a+3d, a+4d, …
The pattern is simple: you start at a, then keep adding d to get the next term.
The common difference d can be positive, negative, or even zero. If d=0, all terms are the same — that's still an AP, just a boring one.
The General Term (nth term)
What if you want the 100th term without writing all 100 numbers? There's a formula.
The first term is a (think of it as a+0⋅d).
The second term is a+d (that's a+1⋅d).
The third term is a+2d.
Notice the pattern: the term number minus 1 tells you how many times d has been added.
So the nth term (also called the general term) is:
Tn=a+(n−1)d
Tn=a+(n−1)d
Example: For the AP 3, 5, 7, 9, ... we have a=3, d=2.
The 10th term: T10=3+(10−1)⋅2=3+18=21.
Why "Arithmetic"?
The name comes from an old property: in an AP, every term (except the first and last) is the arithmetic mean of its neighbours. For three consecutive terms x,y,z in an AP:
y=2x+z
Check: in 3, 5, 7, we have 5=23+7=5. This works for any three consecutive terms.
Sum of the First n Terms
Sometimes you need the total of the first n terms. There's a clever trick.
Write the sum forwards: Sn=a+(a+d)+(a+2d)+⋯+[a+(n−1)d]
Write it backwards: Sn=[a+(n−1)d]+[a+(n−2)d]+⋯+a
Add them term by term. Each pair adds to 2a+(n−1)d, and there are n such pairs. So:
2Sn=n[2a+(n−1)d]
Therefore:
Sn=2n[2a+(n−1)d]
There's another useful form. Since the last term l=a+(n−1)d, we can write:
Sn=2n(a+l)
This is beautiful: the sum of an AP is just the number of terms times the average of the first and last term.
Example: Sum of first 10 terms of 3, 5, 7, ...
S10=210[2⋅3+(10−1)⋅2]=5[6+18]=5×24=120
Quick Reference
| What you need | Formula |
|---|
| nth term | Tn=a+(n−1)d |
| Sum of n terms | Sn=2n[2a+(n−1)d] |
| Sum using last term | Sn=2n(a+l) |
| Common difference | d=Tn+1−Tn |