# Arithmetic Progression | Revision Notes

# Arithmetic Progression

## Important Terms, Definitions and Results

- Some numbers arranged in a definite order, according to a definite rule, are said to form a
**sequence**.
- A sequence is called an
**arithmetic progression** (AP), if the difference of any of its terms and the preceding term is always the same.

i.e., *a*_{n} – a_{n – 1} = constant.
- The constant number is called the c
**ommon difference** of the A.P.
- If
*a* is the first term and *d* the common difference of an AP, then the general form of the AP is *a, a + d, a + 2d, ...*
- Let
*a* be the first term and *d* be the common difference of an AP, then, its *n*th term or general is given by *a*_{n} = a + (n – 1) d
- If
*l* is the last term of the AP, then *n*th term from the end is the nth term of an AP, whose first term is *l* and common difference is *– d*.

∴ *n*th term from the end = Last term + *(n – 1) (– d)*

⇒ *n*th term from the end =* l – (n – 1) d*
- If
*a, b, c*, are in AP, then
- (a + k), (b + k), (c + k) are in AP.
- (a – k), (b – k), (c – k) are in AP.
- ak, bk, ck, are in AP.
- a/k, b/k, c/k are in AP(k ≠ 0).

- Remember the following while working with consecutive terms in an AP.
- Three consecutive terms in an AP.

*a – d, a, a + d *

First term = *a – d*, common difference = *d *

Their sum = *a – d + a + a + d = 3a *
- Four consecutive terms in an AP.

*a – 3d, a – d, a + d, a + 3d*

First term = *a – 3d*, common difference = *2d*

Their sum = *a – 3d + a – d + a + d + a+ 3d = 4a*
- Five consecutive terms in an AP.

*a – 2d, a – d, a, a + d, a + 2d*

First term = *a – 2d,* common difference = *d*

Their sum = *a – 2d + a – d + a + a + d + a + 2d = 5a*

- The sum
*S*_{n} up to *n* terms of an AP whose first term is *a* and common difference *d* is given by

- If the first term and the last term of an AP are
*a* and *l*, then

**a**_{n} = S_{n} – S_{n}_{–1}

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