Class 10 · Chapter 5

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.

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Detailed theory

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 term is

a

Key concepts & formulas

a_n = a + (n-1)d

S_n = \tfrac{n}{2}[2a + (n-1)d] = \tfrac{n}{2}(a+l)

where

l

is the last term.

d = a_n - a_{n-1}

(constant for an AP).

For an odd count

n

, the middle term is

a_{(n+1)/2}

Sum of first

n

natural numbers

= n(n+1)/2

Sum of first

n

odd numbers

= n^2