A sequence is a fundamental arrangement of things, objects, numbers, etc. The sequences may be finite or infinite depending on the number of terms. There are many types of sequences such as geometric sequence, harmonic sequence, and arithmetic sequence which are used to perform the requisite operation. In this topic, we only discuss the arithmetic sequence with its examples.
Arithmetic Sequence
A type of sequence in which a fixed constant is added or subtracted in the preceding terms to generate the subsequent terms is called arithmetic sequence. The arithmetic sequence is formed as
x, x+d, x+2d, x+3d, x+4d……
or
x, x-d, x-2d, x-3d, x-4d……
Where x is the first term and d is the fixed constant also known as common difference. For arithmetic sequence, it is necessary that the common difference between each pair of terms of sequence must be the same.
The two consecutive terms of a sequence are separated by a common difference d which is calculated by subtracting two terms as below:
d = xn+1 – xn
The nth term of an arithmetic sequence is calculated by
xn = x + (n-1)d
Example 1: Conclude that the given numbers are in arithmetic sequence?
7, 11, 15, 19, 23…..
Solution:
We know that if the sequence is arithmetic then the common difference between each pair of terms should be the same. Therefore, we can find the common difference by
d = xn+1 – xn
Put the values
d = 11 – 7 = 4
d = 15 – 11 = 4
d = 19 – 15 = 4
d = 23 – 19 = 4
The common difference d between each pair of terms is 4. So the sequence is arithmetic. Verify the steps and calculations using online arithmetic sequence calculator.
Example 2: Find the common difference to determine whether the sequence is arithmetic or not?
9, 15, 21, 25, 29….
Solution:
The common difference is calculated by
d = xn+1 – xn
Common Difference of first pair of terms
d = 15 – 9 = 6
Common Difference of second pair of terms
d = 21 – 15 = 6
Common Difference of third pair of terms
d = 25 – 21 = 4
Common Difference of fourth pair of terms
d = 29 – 25 = 4
Since the common difference is not the same for all the consecutive terms. Therefore, the sequence is not arithmetic.
Examples to find the nth term of an arithmetic sequence
Example 1: If x = 5 and d = 2 then find the 9th term, 50th term, and nth term of the arithmetic sequence?
Solution:
We use the nth term formula to find the 9th term, 50th term, or any specific term of the arithmetic sequence. The formula for nth term is
xn = x + (n-1)d
Put the given values x = 5 and d = 2 to find the nth term
xn = 5 + (n-1)2
Now, we can find the 9th term by substituting n = 9
x9 = 5 + (9-1)2
x9 = 5 + (8)2
x9 = 5 + 16
x9 = 21
Similarly, the 50th term can be find by substituting n = 50 in the formula of nth term
xn = 5 + (n-1)2
x50= 5 + (50-1)2
x50 = 5 + (49)2
x50= 5 + 98
x50 = 103
Example 2: Find the nth term & 20th term if x1 = 15 and d = 5?
Solution:
As given
x1 = 15
d = 5
Find the nth term by using the formula
xn = x + (n-1)d
Put the values
xn = 15 + (n-1)5
Now, find the 20th term by substituting n=20
x20 = 15 + (20-1)2
x20 = 15 + (19)2
x20 = 15 + 38
x20 = 53
Sum of Arithmetic Sequence
To find the sum of arithmetic sequence if nth term is known, we use the formula
Sn =(n/2) (x1 + xn)
If nth term is not known then we use the formula
Sn =(n/2) (2x + (n-1)d)
Example 1: S = 90 + 110 + 130 + 150…….upto 15th term
Solution:
Identify the values
x = x1 = 90
d = 20
n = 15
In this example, value of nth term is not known so we use the formula
Sn = (n/2) (2x + (n-1)d)
S15 = (15/2) (2(90) + (15-1)(20))
S15 =(15/2) (180 + 460)
S15 = (15/2) (460)
S15 = 3450
To skip the lengthy or manual calculation, you can use an online arithmetic sequence calculator tool to find the solution to your complex problems at once.
Example 2: Find Sum of arithmetic sequence where
x1 = 7, d = 2, xn = 23
Solution:
Given values are
x1 = 7
d = 2
xn = 23
The following formula will be used to calculate the sum of arithmetic sequence
Sn =(n/2) (x1 + xn)
But first of all we will find the nth term by using
xn = x1 + (n-1)d
Put the know values in the formula
23 = 7 + (n-1)2
23 = 7 + 2n – 2
23 = 2n + 5
23 – 5 = 2n
18 = 2n
9 = n
Now, put the all known value in the given formula to calculate the sum of arithmetic progression
Sn =(n/2) (x1 + xn)
Sn = (9/2) (7 + 23)
Sn = (9/2) (30)
Sn = 135
Example 3: Find the sum of the first 15 even numbers?
Solution:
To solve the question, we must know the values of x1, d, x15 and n
As we know, the even number starts from zero as given below
Even numbers = 0, 2, 4, 6, 8, 10, 12……
Where
x1= 0
d = 2-0 = 2
n= 15
To find x15 we use the nth term formula.
xn = x1 + (n-1)d
x15 = 0 + (15-1)2
x15 = 0 + (14)2
x15 = 0 + 28
x15 = 28
Now put all the known values in the formula to calculate the sum of even numbers.
Sn = (n/2) (x1 + x15)
Sn = (15/2) (0 + 28)
Sn = (15/2) (28)
Sn = 420/2
Sn = 210
In algebra, the arithmetic sequence has a linear function because it has a constant difference between each consecutive term. People use different algebraic tools to solve their complicated problems related to the arithmetic sequence.
