Arithmetic Progression
An arithmetic progression has a constant common difference d, with nth term a + (n-1)d and sum n/2 * (2a + (n-1)d).
What is Arithmetic Progression?
An arithmetic progression has a constant common difference d, with nth term a + (n-1)d and sum n/2 * (2a + (n-1)d).
Key formula / rule: The nth term is linear in n and the sum is quadratic in n.
Key points
- Compute the nth term and the sum of an arithmetic progression.
- Identify an arithmetic progression from a described situation.
Common exam trap
Using n as the count of differences rather than the count of terms.
Definitions
- Term
Arithmetic Progression
- Meaning
An arithmetic progression has a constant common difference d, with nth term a + (n-1)d and sum n/2 * (2a + (n-1)d).
- Term
Arithmetic Progression — explanation
- Meaning
Constant additive growth is the simplest pattern in a sequence, and its sum formula follows from pairing terms from the two ends. Most series questions begin by testing whether a sequence is arithmetic.
Learning objectives
Compute the nth term and the sum of an arithmetic progression.
Identify an arithmetic progression from a described situation.
Formulae
- Key point
The nth term is linear in n and the sum is quadratic in n.
- Key point
Three numbers in AP can be written a-d, a, a+d to simplify algebra.
- Key point
The arithmetic mean of two numbers is their average.
Common mistakes
Using n as the count of differences rather than the count of terms.
Applying the sum formula to a sequence whose difference is not constant.
Keywords
Arithmetic
Progression
Practice preview
The sequence 5, 12, 19, 26, ... is an Arithmetic Progression. What is its common difference?…
easy
Find the sum of the first 10 terms of an AP whose first term is 2 and common difference is 3.…
easy
Which term of the AP 3, 8, 13, 18, ... is 78?…
medium