### Arithmetic Sequences and Series | Arithmetic Mean

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:

an = a + (n – 1)d

nth term from the last term

an = l – (n – 1)d

Sum of an terms A.P

Sn = n2[2a + (n – 1)d]

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

Sn = n2(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: T25 = 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 a1, a2, a3………… an is given by

= a1 + a2 + a3 +…………+ ann

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 + 586
= 3756
= 62.5 km/h

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