Formula for Arithmetic Progression (AP)

Here you you will learn what is arithmetic progression (AP) and formula for arithmetic progression.

Let’s begin –

Arithmetic Progression (AP)

A sequence is called an arithmetic progression if the difference of a term and the previous term is always same i.e.

\(a_{n+1}\) – \(a_n\) = constant (=d) for all n \(\in\) N

The constant difference generally denoted by d is called the common difference.

for example : 1, 4, 7, 10 ….. is an AP whose first term is 1 and the common difference is equal to 4 – 1 = 3.

Also Read : Formula for Geometric Progression (GP)

Formula for Arithmetic Progression

(a) General term of an AP ( nth term of ap)

Let a be the first term and d be the common difference of an AP. Then its nth term or general term is a + (n – 1)d

i.e. \(a_n\) = a + (n – 1)d.

(b) nth term of an AP from the end

Let a be the first term and d be the common difference of an AP having m terms. Then nth term from the end is \((m – n + 1)^{th}\) term from the beginning.

\(\therefore\) nth term from the end = \(a_{m-n+1}\)

= a + (m-n+1-1)d = a + (m-n)d

Also nth term from the end = \(a_m\) + (n-1)(-d)

[\(\because\) Taking \(a_m\) as the first term and the common difference equal to ‘-d’ ]

(c) Sum to n terms of an AP

The sum \(S_n\) of n terms of an AP with first term ‘a’ and common difference ‘d’ is

\(S_n\) = \(n\over 2\) [2a + (n-1)d]

or, \(S_n\) = \(n\over 2\) [a + l] , where l = last term = a + (n-1)d

Example : Show that the sequence 9, 12, 15, 18, ……. is an AP. find its 16th term, general term sum of first 20 terms.

Solution : We have, (12 – 9) = (15 – 12) = (18 – 15) = 3. Therefore, the given sequence is an AP with the common difference 3.

first term = a = 9

\(\therefore\) 16th term = \(a_{16}\) = a + (16-1)d

\(\implies\) \(a_{16}\) = 9 + 15*3 = 54

\(\because\) General term = nth term = \(a_n\) = a + (n-1)d

\(\therefore\) \(a_n\) = 9 + (n-1)*3 = 3n + 6

Now, sum of first 20 terms = \(S_{20}\) = \(20\over 2\) [2*9 + (20-1)3]

\(S_{20}\) = 10[18 + 19*3]

= 750