Class 10 · Chapter 5 · Mathematics

Arithmetic Progressions Cheat Sheet

Formulae

General term (nth term) of an AP

a_{n} = a + (n-1)d

a = first term, d = common difference, n = term number.

Common difference

d = a_{n} - a_{n-1}

The constant difference between any term and the term just before it.

Sum of first n terms

S_{n} = \frac{n}{2}\left[2a + (n-1)d\right]

Used when the first term and common difference are known.

Sum using first and last term

S_{n} = \frac{n}{2}(a + l)

a = first term, l = last (nth) term.

nth term from the sum

a_{n} = S_{n} - S_{n-1}

The nth term equals the sum of n terms minus the sum of the first n − 1 terms.

Three terms in AP

2b = a + c

If a, b, c are in AP, the middle term is the average of the other two.

nth term from the end

l - (n-1)d

Counting backwards from the last term l with common difference d.

Sequence: An ordered list of numbers, each called a term, arranged according to a rule.
Arithmetic progression (AP): A sequence in which the difference between consecutive terms is constant.
Common difference: The fixed number added to each term to get the next term; denoted d.
First term: The initial term of the progression; denoted a.
General term: The expression aₙ = a + (n − 1)d giving the nth term of an AP.
Sum of n terms: The total of the first n terms of an AP; denoted Sₙ.

In an AP, the difference between any two consecutive terms is always the same.
The nth term of an AP is linear in n: aₙ = a + (n − 1)d.
The sum of the first n terms of an AP is Sₙ = n/2 [2a + (n − 1)d].
If the last term l is known, Sₙ = n/2 (a + l).
A sequence is an AP if aₙ₊₁ − aₙ is a constant for every n.
The graph of the terms of an AP lies on a straight line.
When a finite number of terms is taken, the common difference can be found from any two consecutive terms.
APs are used to model situations with a constant increase or decrease, such as savings and salaries.