In this article, we will be discussing Arithmetic progressions formulas & then we will look at some Arithmetic progression solved example problems.
Concept of Arithmetic Progression
An arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 3, 5, 7, 9, 11, 13, 15 … is an arithmetic progression with common difference of 2.
A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression. The sum of a finite arithmetic progression is called an arithmetic sum.
The behavior of the arithmetic progression depends on the common difference d. If the common difference is:
- Positive, the members (terms) will grow towards positive infinity.
- Negative, the members (terms) will grow towards negative infinity.
Example: Let’s check whether the given sequence is A.P: 1, 3, 5, 7, 9, 1. To check if the given sequence is A.P or not, we must first prove that the difference between the consecutive terms is constant. So, d = a2– a1 should be equal to a3– a2 and so on… Here,
d = 3 – 1 = 2 equal to 5 – 3 = 2
Real-time Example: Suppose while returning from school, you get into the taxi. Once you ride a taxi you will be charged an initial rate. But then the charge will be per mile or per kilometer. This show that the arithmetic sequence for every kilometer you will be charged a certain constant rate plus the initial rate. To understand this let us study the topic of arithmetic progression in detail.
Arithmetic Progression Formulas
Here are some of the important Arithmetic Progression related formulas:
- The general form of an Arithmetic Progression is a, a + d, a + 2d, a + 3d and so on.
- The nth term of an Arithmetic Progression series is An = a1 + (n - 1) d, where An = nth term and a1 = first term. Here d = common difference = An - An-1.
- The sum of the first n terms of an Arithmetic Progression series is S =(n/2)[2a + (n- 1)d]
- The sum of n terms can be calculated using the below given formula if the last term is given,
- Also, An = Sn - Sn-1 , where An = nth term
Arithmetic Progression Problems
Example 1: Find the 15th term of the arithmetic progression 3, 9, 15, 21,....?
In the given AP,
we have a = 3, d = (9 - 3) = 6, n =15
T15 = a + (n - 1)d
= 3 + (15 -1)6
= 3 + 84 = 87