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• Posted on May 20th, 2019 in Mathematics
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## Sequence

A succession of numbers formed and arranged in a definite order according to a certain definite rule is called a sequence.Continue Reading Aptitude – Progression – Mathematics Notes – For W.B.C.S. Examination.

## Arithmetic Progression (A.P.)

It is a sequence in which each term, except the first one differs the preceding term by a constant. This constant is called the common difference. We denote the first term by a, common difference by d, nth term by Tṇ and the sum of first n terms by Sṇ.

### Examples

```5, 8,11,14,17...is an A.P. in which a=5 and d = (8-5) =3.
8, 5, 2,-1,-4,-7.... is an A.P. in which a = 8 and d = (5-8) = -3.
```

## General Term of an A.P.

In a given A.P., let first term =a, common difference =d. Then,

```Tn= a + (n-1) d.
Sum of n terms of an A.P.
Sn = n/2[2a+ (n-1) d]
Sn = n/2 (a + L), where L is the last term.
```

## Geometrical Progression (G.P.)

A sequence in which each term, except the first one bears a constant ratio with its preceding term, is called a geometrical progression, written as G.P. The constant ratio is called the common ratio of the G.P. We denote its first term by a and common ratio by r.

### Example

```2, 6, 18, 54, is a G.P.in which a=2 and r=6/2=3.
24, 12, 6, 3... Is a G.P. in which a = 24 and r = 12/24=1/2.
```

General Term of a G.P.: In a G.P. we have

```Tn= arn-1
Sum of n terms of a G.P.
Sn = a (1-rn)/ (1-r), When r < 1
a (r - 1n)/(r-1), When r > 1
```

## Arithmetic Mean

```A.M. of a and b = 1/2(a+b).
```

## Geometric Mean

```G.M. of a and b =√ab
```

## Some General Series

```(i) 1+2+3+4+…….+n=1/2n (n+1).
(ii) 12+22+32+42+……+n2 = n(n+1)(2n+1)/6
(iii)  13+23+33+43+…..+n3= {1/2 n(n+1)}2

``` ` `