Arithmetic progression: An introduction with examples
A sequence is a set of events that follow a logical and predictable pattern. In mathematics and in nature we look at different patterns that enhance the beauty for the viewers.
Everything in the world fellow a logical pattern like roses, age, time, the moment of the sun, and starts. In this article, we will discuss about a special type of sequence which has some proper properties and applications in real life.
What is an Arithmetic Sequence?
An arithmetic sequence is a sequence in which every two consecutive terms have the same common difference. It means that the next term in this sequence will be obtained by adding the common difference in the given term. The arithmetic sequence is also known as arithmetic progression.
Formula of Arithmetic Sequence
The general formula of A.P is
an = a1 + (n -1) d
In the formula,
d is the common difference between any two successive terms.
The nth term is represented by an.
a1 is denote the first term.
sum of the A.P terms Sn.
The Sum of big data values is quite difficult. So you can reduce the difficulty of calculating larger terms with the help of an arithmetic sequence calculator to get the result according to the given formula.
Let us have an A.P whose terms are a1, a2, a3… an.
Then a1 is known as the first term of the A.P.
Let us have the terms 121, 221, 321, 421…
It is an A.P with having the first term 121.
The common difference (d) is defined as the difference between any two successive terms of the A.P. We have a general formula for common difference d = an-an-1.
Let us have an A.P 10,25,40,55…
d = 25 – 10 = 15
d = 40 -25 = 15
Note: Common differences remain constant if the sequence is an A.P.
The sequence in which we have the last term is known as the nth term of the sequence. The nth term is represented by an.
Let us have an A.P 10, 20, 30… 90
The given A.P has a common difference d = 20 – 10 =10
The nth term means the last or final term or terminal term is an = 90
Number of the term
The terms of a sequence are arranged in a well-ordered way. Each term is associated with a specific number. 1st, 2nd, 3rd, … etc.
If we have the nth term of an A.P is 50, the first term is 10 the and common difference is d= 7. Find which term is 50.
Step 1: Write the given data in standard notations
an = 150
a1 = 10
d = 7
Step 2: The formula of the A.P is
an = a1 + (n -1) d
Step 3: Put the values of know terms in the formula we have,
150 = 10 + (n – 1) × 7
150 – 10 = (n – 1) × 7
140 = (n – 1) × 7
140 / 7 = (n -1)
140/7 +1 = n
(140 + 7)/7 = n
147/7 = n
21 = n
n = 21
So the given term 50 is the 21st term of the given sequence.
Examples of Arithmetic Sequence
To understand the concept of A.P, you have to solve some practical examples to illustrate the basics of A.P. We discuss each example in detail with a step-by-step solution.
Dr. Asad Gohar saved $ 10 in the first week of a year and then increased his weekly saving by $ 1.75. If his weekly savings reach $ 20.74 in the nth week. Then find the number of weeks.
Step 1: Find the A.P from the given statement
Dr. Asad Gohar saving on the first week = $ 5
ON 2nd-week saving = 10 + 1.75 = $ 11.75
ON 3rd- week saving = 11.75 + 1.75 = $ 13.50
Hence the A.P of the saving is
10, 11.75, 13.50…
a = 10
d = 11.75 – 10 = 1.75
Dr. Asad Gohar’s savings for the nth week = an = $ 20.75
Step 3: The formula of A.P for the nth term
an = a1 + (n -1) d
20.75 = 10+ (n -1) *1.75
20.75 = 10 + 1.75n -1.75
20.75 = 1.75n + 8.25
20.75 – 8.25 = 1.75n
12.50 = 1.75 n
17.50 / 1.75 = n
8 = n
In the 8th week, his saving is $ 20.75.
Find the sum of the first seven terms if the common difference is 10 and the first term is 5?
Step 1: Write the data in the form of mathematical notations
Total number of values = 7
First-term = a1 = 5
Common difference = 10
Step 2: Formula of the sum of A.P
Sn = n/2 + (2a1 + (n-1) d)
Step 3: Put the corresponding values in the formula
S7 = 7/2+ (2*5+ (7 - 1) * 10)
S7 = 3.5+ (10+6*10)
S7 = 3.5 + (10 +60)
S7 = 3.5 + 70
S7 = 73.5
Hence the sum of the first 7 terms is 73.5.
Application of arithmetic progression in real life
Sunflower petals and how they grow
Scales from a pineapple (the rows and their slopes, the number of scales in each row exhibiting the pattern)
Sides of flat bananas
The majority of tree branches (the way they spread out)
The majority of flower petals
Waves are generated by hair growing from the back of our heads.
In this post, we have learned about the Arithmetic progression and its applications in daily life. You can understand the type of given sequence.
You can calculate missing terms and the sum of any number of the terms which is difficult to add with usual methods.