Arithmetic Progressions

**Category : **10th Class

**Arithmetic Progressions**

**Sequence:**Numbers arranged in a definite order according to definite rule are said to be in a sequence.

**Term:**Each number of a sequence is called a term.

**\[{{\mathbf{n}}^{\mathbf{th}}}\]term:**The term occurring at the\[{{n}^{th}}\]place of a sequence is called its n"1 term, usually denoted by\[{{t}_{n}}\].

**Progressions:**Sequences that follow a definite pattern are called progressions.

**Arithmetic progressions:**sequence in which each term differs from its preceding term by a fixed number (constant) is called an arithmetic progression, denoted as A.P.

**Common Difference:**The fixed number by which any two successive terms of an A.P. differ is called the common difference of A.P. denoted by 'd'. So,\[\text{d}={{\text{t}}_{n}}-{{t}_{n-1}}\].

An A.P. of 'n' terms with first term 'a' and common difference 'd' is a, a +d,...a +(n- 1)d.

**Arithmetic series:**A series obtained by adding the terms of an A.P. is called an arithmetic series.

**The general term (\[{{\mathbf{n}}^{\mathbf{th}}}\]term) of an A.P.:**If the first term of an A.P. is 'a' and the common

- difference is 'd', then its n111 term is given by\[{{t}_{n}}=a+(n-1)d\].

**The general term from the end of an A.P.:**If 'a' is the first term, 'd' the common difference and \['l'\]the last term of a given A.P., then its \[{{n}^{th}}\]term from the end is \[l-(n-1)d\].

**Selection of term of an A.P.:**Terms of an A.P. must be selected in such a way, that on taking the sum of the terms, one unknown is eliminated automatically.

(a)To select three terms of an A.P. with common difference 'd', choose a - d, a, a + d.

(b) To select four terms of an A.P. with common difference 2d, choose a - 3d, a - d, a + d, a + 3d.

- (c) To select five terms of an A.P. with common difference d, choose a - 2d, a - d, a, a + d, a + 2d.

(d) To select six terms of an A.P. with common difference 2d, choose \[\text{a}-\text{5d},\text{ a}-\text{3d}\], \[\text{a}-\text{d},\text{ a},\text{ a}+\text{d},\text{ a}+\text{3d},\text{ a}+\text{5d}\]

**The sum to 'n' terms of an A.P.:**The sum of first 'n' terms of an A.P. is given by \[S=\frac{n}{2}[2a+(n-1)d]\], where 'a' is the first term and 'd' is the common difference.

**Arithmetic Mean:**If a, A and b are in A.P., then A is said to be the arithmetic mean (A.M.) between a and b. The arithmetic mean between two numbers 'a' and 'b' is given by\[(a+b)/2\].

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