An arithmetic sequence is a sequence of numbers such that the difference of any two consecutive terms of the sequence is a constant. Its also called **Arithmetic Progression** and denoted as **A.P**

A.P = {a, a+d, a+2d, a+3d, ….,a+(n-1)d,….}

where:

a is the first term, and

d is the difference between the terms (called the “common difference”)

Successive term – Preceding term = constant

**Example:** 2,4,6,8,10….

In the above sequence with the common difference 2.

If the first term of an arithmetic sequence is a1 and the common difference is d, then the nth term of the sequence is given by:

a_{n} = a + (n – 1)d

nth term from the last term

a_{n} = l – (n – 1)d

Sum of a_{n} terms A.P

S_{n} = *n*2[2a + (n – 1)d]

If l is the last term i.e nth term of A.P then

S_{n} = *n*2(a + l)

**Problem 1**: If 4,7,10,13,16,19,22……is a sequence, Find:

Common difference

nth term

25st term

**Solution**: Given sequence is, 5, 10, 15, 20, 25, 30…..

a) The common difference = 10 – 5 = 5

b) The nth term of the arithmetic sequence is denoted by the term Tn and is given by **Tn = a + (n-1)d**, where “a” is the first term and d, is the common difference.

c) 21st term as: T_{25} = 4 + (25-1)5 = 4 + 24 x 5 = 125.

**Arithmetic Mean**

In general language arithmetic mean is same as the average of data. If three numbers a, b, c are in A.P then one (b) is called the arithmetic mean between extremes (a and c).

b = *(a + c)*2

Mean = *Sum of all observation *number of observation

The mean of n observation a_{1}, a_{2}, a_{3}………… a_{n} is given by

= *a _{1} + a_{2} + a_{3} +…………+ a_{n}*n

Where the symbol ∑ called sigma which stands for summation.

**Problem**: Find the mean driving speed for 6 different bikes.

66 km/h, 57 km/h, 71 km/h, 54 km/h, 69 km/h, 58 km/h

**Solution**: number of observation = 6

Mean = *Sum of all observation *number of observation

= *66 + 57 + 71 + 54 + 69 + 58*6

= *375*6

= 62.5 km/h

I hope, this article will help you a lot to understand the **Arithmetic Sequences and Series | Arithmetic Mean**. If you still have any doubts and problems with any topic of mathematics you can ask your problem in the Ask Question section. You will get a reply shortly.

**Related Topic:**

Sequence and Series | Progression | Type of Sequence

Geometric Sequences and Series | Geometric Mean

Harmonic Sequence | Harmonic Mean

Number Series Concept And Tricks | Problems