# Maths formulas for class 10 chapter- Arithmetic Progressions Formula

#### Math Formulas

## Class 10 chapter- Arithmetic Progressions Formula

### Arithmatic Progression

A succession of numbers is called a **“ Sequence”** or “Sequence” means an arrangement of number in a definite order according to some rule.

**Finite sequence** →Sequence containing finite number of terms is finite sequence

**Infinite sequence** →Sequence containing infinite number of terms is infinite sequence.

The sequence in which every term except the first obtained by adding fixed number (Positive or Negative) to the preceding terms is called **“ Arithmetic Progression”** (A.P.)

A progression is a type of sequence for which it is possible to obtain a formula for the nth term. The Arithmetic Progression or we can say AP is the most commonly used sequence in maths with easy-to-understand formulas.

an = a + (n − 1) × d

where,

a_{n }= n^{th }term

a = firsr tem

d = common difference

n = no. of terms

Thus any sequence a, a_{2}, a_{3}, ……, a-n- is called an Arithmetic Progression (A.P.) if, a_{n+1} = a_{n} + d, n∈ N; where a is called the first term and the constant number d is called the common difference of the A.P.

### The nth term or general term of an A.P.

Let us consider an A.P. with first term a and common difference d i.e. a, a + d, a + 2d, a + 3d … then

1^{st} term= a_{1}= a=a+(1-1)

2^{nd} term= a_{2}= a+d=a+(2-1)d

3^{rd} term = a_{3} = a+2d = a+(3-1)d

**nth term→ a _{n} = a + (n – 1)d**

### Properties of an A.P.

- If a constant is added to each term of an A.P. the resulting sequence is also an A.P.
- If a constant is subtracted from each term of an A.P. the resulting sequence is also an A.P.
- If each term of an A.P. is multiplied by a constant, then the resulting sequence is also an A.P.
- If each term of an A.P. is divided by a non zero constant then the resulting sequence is also A.P.

### The sum of n terms of an A.P

German mathematician Carl Friedrich gauss derive the method of finding sum of n term of an A.P.

Let S_{n} be the sum of n terms of the A.P. then

The sequence in which every term except the first term bears a constant ratio to the term immediately preceding it, is called **“ Geometric Progression”** (G.P.)

A sequence a, a_{2}, a_{3}, …… a_{n} is called Geometric progression (G.P.) if, a_{n+1}/a_{n} = r (constant) for every k ≥1.

The Geometric progression is a,ar,ar^{2},ar^{3}.......is called the first term and the constant number r ¹ 0 is called the common ratio of the G.P.

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