What is Arithmetic sequence? Explained with Examples
In algebra, we often see numbers given in specific patterns. Some of them are easy to understand but some of them are logically difficult. A collection of numbers given in a specific pattern is termed a sequence.
In sequence, numbers are given in different orders, and in some cases; we have to find the next terms that follow the given pattern. By this technique, we can judge in what directions the numbers will move. This technique is helpful in many problems.
In this article, we will study the arithmetic sequence, its formula, and how to find the sum of the nth term of the arithmetic sequence along with their examples.
Arithmetic Sequence
We can define an arithmetic sequence in two different ways.
- An arithmetic sequence is one in which the difference between every two successive terms is equal.
- In any sequence, every term is obtained by adding a fixed number (zero negative or positive integer) to its previous term.
To understand an Arithmetic sequence, let us check an example of it. Check 1, 6, 11, 16……. is it an arithmetic sequence or not?
It can be seen that every number in the given sequence is obtained by adding 5 to its previous term, so its common difference is the same. Therefore, we can say that 1, 6, 11, 16……. is an arithmetic sequence.
Formulas
We have two arithmetic sequence formulas.
- The following formula may be used to determine the nth term in an arithmetic sequence:
an = a1 +(n-1)d
or,
an = an-1 +d
Where ‘d’ is the common difference d= an – an-1
- The formula for calculating the sum of an arithmetic sequence's first n terms.
Sn = (n/2) [2a + (n - 1)d]
If we want to find any term in the arithmetic sequence then we can use the arithmetic sequence formulas.
The recursive formula of an Arithmetic Progression
The recursive formula of the Arithmetic progression is given as follows:
A.P =a, a+d, a+2d, a+3d ......up to n terms.
Nth Term of Arithmetic Sequence
Let a1, a2, a3…. be an arithmetic sequence. Its nth term is given by an = a1 +(n-1)d. This is also termed the general form of the arithmetic sequence.
Important Notes on Arithmetic Sequence:
- The difference between each pair of succeeding integers in an arithmetic series is always the same.
- The common difference of an arithmetic sequence a1, a2, a3, ... is, d = a2 - a1 = a3 - a2 = ...
- The nth term of an arithmetic sequence is an = a1 + (n−1)d.
- The sum of the first n terms of an arithmetic sequence is Sn = n/2[2a1 + (n − 1)d].
- There are three possible values for the common difference of arithmetic sequences: positive, negative, or zero.
How to solve the problems of arithmetic sequence?
Example 1:
Consider the sequence 4, 8, 12, 16, 20….. is an arithmetic sequence because every term is obtained by adding a constant number 4 to its previous term. Find the next 5 terms of the sequence.
Solution:
Given data:
a1=4, a2=8,a3=12, a4=16, a5=20
d = an – an-1 = 8 – 4 = 12 – 8 = 16 – 12 = 20 – 16 = 4
To find:
a6 =?, a7 =?, a8=?, a9=?, a10=?
Formula:
an = an-1 +d
a6= a6-1+d = a5+d= 20+4 = 24
a7= a7-1+d = a6+d = 24+4 = 28
a8= a8-1+d = a7+d = 28+4 = 32
a9= a9-1+d = a8+d = 32+4 = 36
a10= a10-1+d = a9+d = 36+4 = 40
Example 2:
Find a17 of an arithmetic sequence if a15 = – 72 and d = 7.
Solution:
By using the recursive formula,
a16 = a15 + d = – 72 + 7 = – 65
a17 = a16 + d = – 65 + 7 = – 58
Therefore, a17 = – 58
An nth term calculator could be used to solve arithmetic sequence problems easily with a step-by-step solution.
Example 3:
Mr. Hassan works in a well-renowned organization. He earns Rs 120,000 per annum and his salary increases Rs 20,000 per annum. Then how much does he earn at the end of the first five years?
Solution:
The amount earned by Mr. Hassan for the first year is a = 120,000. The increment per annum is, d = 20,000. We have to calculate his earnings in the first 5 years. Hence n = 5. Substituting these values in the sum of arithmetic sequence formula, we have
Sn = (n/2) [2a + (n –1)d ]
Sn = (5/2) [2(120000) + (5– 1)(20000)]
Sn = (5/2) [240000+4(20000)]
Sn= (5/2) [240000+80000]
Sn= (5/2) [320000]
Sn= 800,000
He will earn Rs 800,000 in 5 years.
Example 4:
Extract the formula of a sequence with two given terms, a5= – 13 and a18=65
Solution:
Given data:
a5= – 13
a18=65
Formula:
an = a1 +(n– 1)d
For n=5 as a5= – 13
a5=a1 + (5– 1)d
– 13=a1+4d ← Equation#1
For n= 18 as a18=65
a18= a1+ (18– 1)d
65= a1+17d ← Equation#2
Solving a system of equations, we have
– 13=a1+4d ← Equation#1
65= a1+17d ← Equation#2
Using the method of equating the coefficient of linear equations
– 13=a1+4d
±65= ±a1±17d
– 78 = – 13d
d= – 78/ – 13
d= 6
Using d=6 in Equation#1, we have
– 13=a1+4(6)
– 13= a1+24
– 13– 24=a1
a1= – 37
Since a1= – 37 and d=6, then formula will be
an = a1 +(n– 1)d
an= – 37 + (n– 1)6
an= – 37 + 6n– 6
an= 6n– 43
Applications
Arithmetic sequences have numerous applications in various fields, ranging from mathematics and physics to everyday life scenarios. Here are some practical applications of arithmetic sequences:
Financial Planning: Arithmetic sequences are used in financial planning, budgeting, and investment scenarios. For instance, when setting up a savings plan with regular contributions, the amount saved each month forms an arithmetic sequence. The knowledge of arithmetic sequences helps in projecting future savings and estimating the growth of investments.
Computer Programming: In computer science and programming, arithmetic sequences are often used for generating sequences of numbers or loop iterations.
Construction and Engineering: In construction projects, arithmetic sequences are used to evenly space components, such as building pillars or floor tiles, at equal intervals.
Physical Sciences: In physics, arithmetic sequences are used to describe uniform motion, where the position of an object changes by the same amount over equal intervals of time. They are also applied in kinematics to analyze linear motion.
Conclusion
In this article, we have studied an arithmetic sequence along with its useful terms. The formula used to calculate the nth term in the arithmetic sequence can help us find any term in the sequence. Through its examples, we can solve all the problems relevant to arithmetic sequences.