01 · THE BASICS
What is an Arithmetic Progression (AP)?
An Arithmetic Progression (AP) is a list of numbers in which each term is obtained by adding a fixed number d to the preceding term except the first term a.
General Form of an AP: a, a + d, a + 2d, a + 3d, ...
Common Difference (d)
The fixed number d is called the common difference.
d = ak - ak-1 (It can be positive, negative, or zero!).
02 · EXAMPLES
Examples of AP Sequences
Positive d
1, 4, 7, 10, 13...
First term a = 1, Common difference d = +3Negative d
100, 70, 40, 10, -20...
First term a = 100, Common difference d = -30Zero d
3, 3, 3, 3, 3...
First term a = 3, Common difference d = 003 · SOLVED EXAMPLE
Checking if a Sequence is an AP
EXAMPLE 1
Which of the following forms an AP? (i) 2, 4, 8, 16... (ii) 2, 5/2, 3, 7/2...
(i) For 2, 4, 8, 16...:
a₂ - a₁ = 4 - 2 = 2
a₃ - a₂ = 8 - 4 = 4
a₃ - a₂ = 8 - 4 = 4
Since a₂ - a₁ ≠ a₃ - a₂, the common difference is not constant. Not an AP!
(ii) For 2, 5/2, 3, 7/2...:
a₂ - a₁ = 5/2 - 2 = 1/2
a₃ - a₂ = 3 - 5/2 = 1/2
a₄ - a₃ = 7/2 - 3 = 1/2
a₃ - a₂ = 3 - 5/2 = 1/2
a₄ - a₃ = 7/2 - 3 = 1/2
Since ak+1 - ak is constant (1/2), this is an AP with a = 2, d = 1/2.
04 · QUICK SUMMARY
Key Takeaways
01
An AP is uniquely determined if you know the first term a and common difference d.
02
Common difference formula: d = a₂ - a₁ = a₃ - a₂ = ... = ak+1 - ak.
03
Finite AP has a last term; Infinite AP continues indefinitely.