# Arithmetical Progression (AP)

## Arithmetical Progression (AP)

A sequence is said to be an arithmetical progression, if the difference of a term and its previous term is always same.

i.e., an₊₁ – an = Constant = d,  ∀ n ϵ N

The constant difference, generally denoted by d is called the common different

(or)

An athematic progression (AP) is a sequence whose terms increase or decrease by a fixed number. The fixed number is called the common difference of the AP.

In other words, if a₁, a₂, … an are in the AP, then

a₂ – a₁ = a₃ – a₂ = … = an – a n ₊ ₁ = d

If a is the first term and d is the common difference, then AP can be written as

a, a + d, a + 2d, … [a + (n – 1) d]

Examples:

(i) 1, 4, 6, 8 …

(ii) 3, 5, 7 …

The nth term of an AP: Let a be the first term d be the common difference and ‘l’ be the last term of AP, then nth term is given by

Tn = l = a + (n – 1) d

Where d = Tn – Tn ₊ ₁

The nth term from last is T’n = l – (n – 1) d

The sum of n term of an AP: Suppose there are n terms of a sequence, whose first term is a, common difference is d and last term is l, then sum of n terms is given by

Sn = n/2[2a + (n – 1) d] = n/2[a + l]

Example: If the roots of the equation x³ – 12x² + 39 x – 28 = 0 are in AP, then their common difference will be

Solution: Given that,

x³ – 12x² + 39 x – 28 = 0

Since the given equation is cubic, therefore we take three roots

Let the roots be a – d, a, a + d

Sum of their number in AP = a – d + a + a + d = 12

⇒ 3a = 12

a = 4

The given equation x³ – 12x² + 39 x – 28 = 0 can be written as

(x – 4) (x² – 8x + 7) = 0

x = 1, 4, 7 (or) 7, 4, 1

d = ± 3.