A progression is a sequence of numbers, for which we can find the Nth term (any term in the sequence) using a formula. We can easily predict the numbers in a progression. In mathematics, there are three different types of progression. They are Arithmetic progression, Geometric progression, and Harmonic Progression. In this article, we look at the definition, formula, and examples of arithmetic progression.
Arithmetic progression
Arithmetic progression definition
Arithmetic sequence or progression is a mathematical sequence in which the difference between two consecutive terms is always a constant. It’s abbreviated as AP. For example in the sequence 3,9,15,21,27,33,39,45…… the difference between two consecutive numbers is a constant value of 6. Hence it is an arithmetic progression.
Common difference
The difference between the two consecutive numbers is called a common difference and is denoted by ‘d’. The value of d should be fixed in the entire sequence. If in case even for a single case if the value of d is not constant then the sequence is not an AP.
Arithmetic progression examples
Imagine the heights of a stair-case in centimeters – 100, 200, 300, 400,500….. We can see it has a common difference of 100.
Marks obtained by a student in consecutive examinations – 7, 20, 33, 46, 59, 72. This is an AP with 6 terms with the first term as 7, a common difference of 13, and the last term 72.
The distance covered by a vehicle after every hour in kilometers(under uniform motion) – 0, 50, 100, 150, 200…
The goals scored by a football team in 4 consecutive matches- 1, 2, 3, 4
Visitors in a theme park on consecutive days – 20 thousand, 15 thousand, 10 thousand, 5 thousand… Here the common difference is -5000.
Examples of progression in images
The above image shows an image of rassberry in2 dimension. Here we can see the arithmetic progression of the patter: 1, 2, 3, 4, 5
Here in the above example, the height of the pencils are in descending order with a common difference, hence this is an example of arithmetic progression.
The company with a constant growth rate year on year can be considered as an example of an arithmetic sequence.
This can be considered somewhat like an example of arithmetic progression in chemistry. The number of benzene rings in each row is in arithmetic progression with a common difference of 1. The arithmetic progression is 1,2,3.
Arithmetic sequence formula
If we want to find any term ( nth term ) in the arithmetic sequence formula should help you to do so. It is important to find the exact known Values from the problem that will eventually be substituted into the formula itself.
Arithmetic progression formula for the nth term:
To find out the nth term of the sequence,
A n = a 1 +(n- 1) d
Where,
a n is the term that we have to find
a 1 = first term in the sequence
n is the nth term.
d is the common difference of any pair of consecutive or adjacent numbers
Sum formula for an arithmetic progression
Consider an arithmetic progression, whose first term is a 1 or a and the common difference is d.
•The sum of first n terms of an arithmetic progression when the nth term is not known is :
S n = n/2 [ 2a + (n-1)d]
•The sum of first n terms of an arithmetic progression when the nth term, a n is known as :
S n = n/2[a 1 + a n ]
Solved numericals on arithmetic progression
1 .Find out the value of n if,
In an AP , a =10, d = -1 and a(n) = -20.
From the given statement,
a= 10,
d = -1
a(n)= -20
Applying the nth term formula for arithmetic progression,
a (n) = a +( n-1) d
-20 = 10 + (n-1) (-1)
– 20= 11 – n
N = 31
That is, the number of terms in this Arithmetic progression, n =18.
2. Find the sum of the first 30 multiples of 3.
From the given statement ,
a = 3 ,
n =30
d = 3
Applying the sum formula for arithmetic progression,
S = n/2 [2a + (n – 1) × d]
= 30/2[2 (3) + (30 – 1) × 3]
= 15[6 + 87]
S = 1395
3. Findout if the given sequence is arithmetic:5, 23, 40, 57, 73, 90….
For a given sequence to be arithmetic the difference between consecutive terms must be common.
- d1 = 17
- d2 = 17
- d3 = 17
- d4 = 16
- d5 = 17
All the differences are not equal, hence this sequence cannot be called an arithmetic progression.
4. Find out if the heights of the skyscrapers are in arithmetic progression or not?
There is no common difference between the height of consecutive building, hence this is not an example of arithmetic progression.
