# 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

### Related Questions

If 100 times the 100th term of an AP with non-zero common difference equal to the 50 times its 50th term, then the 150th term of AP is

If x, y and z are in AP and $$tan^{-1}x$$, $$tan^{-1}y$$ and $$tan^{-1}z$$ are also in AP, then