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\[(a+b)/2\].
