CBSE · Class 6-12

Class 10 · Mathematics (Standard)

Unit 5 · Class 10Mathematics (Standard)

Arithmetic Progressions

Sequences where each term differs from the previous by a fixed common difference; formulas for the $n$-th term and sum of the first $n$ terms.

Lessons

6 concepts

1
Definition and Notation
An Arithmetic Progression (AP) is a sequence where each term after the first is obtained by adding a fixed number $d$ (the common difference) to the previous term. Standard form: $a, a+d, a+2d, a+3d, \ldots$ . The first t…
2
General Term ($n$-th term)
The $n$ -th term is $a_n = a + (n-1)d$ . Use this to find any term, or to check membership: given a value $x$ , solve $x = a + (n-1)d$ for $n$ ; if $n$ is a positive integer, $x$ belongs to the AP.
3
Sum of First n Terms
$S_n = \dfrac{n}{2}\big(2a + (n-1)d\big)$ . If the first term $a$ and last term $l$ are known, $S_n = \dfrac{n}{2}(a + l)$ . To find how many terms sum to a target $T$ , solve $S_n = T$ (a quadratic in $n$ ).
4
Selecting Terms Symmetrically
For problems with sums of consecutive AP terms, take them symmetrically to simplify. Three terms: $a-d, a, a+d$ . Four terms: $a-3d, a-d, a+d, a+3d$ . Five terms: $a-2d, a-d, a, a+d, a+2d$ . This makes the sum simply $ka$ f…
5
AP in Real Life
APs model salaries with annual fixed hikes, depreciation, seat-count in rows of an auditorium, savings by depositing a fixed amount each period, distance covered by falling objects (with gravity). Set up $a, d$ carefully…
6
Sums via Formulas
Two special sums: (i) first $n$ natural numbers $= n(n+1)/2$ ; (ii) first $n$ odd numbers $= n^2$ ; (iii) first $n$ multiples of $k$ is $k\cdot n(n+1)/2$ . Recognise these to save time.

Skills to practise

Skill · medium

Arithmetic Progressions — Practice