What is an arithmetic sequence and how to solve the problems by using its formulas?
In mathematics, the arithmetic sequence is used widely in sequences. The sequence in which differences among two successive numbers are the same is known as an arithmetic sequence. The ordered group or set of numbers are usually known as sequences.
The distance among the numbers of an arithmetic sequence is constant. For example, 2, 7, 12, 17, 22, 27, … is a sequence having the same difference among every two numbers that is five so we can conclude that the given sequence is an arithmetic sequence.
What is an arithmetic sequence?
A sequence of numbers having constant distance among the numbers is known as the arithmetic sequence. In simple words, the terms of the sequence can be made by adding the constant value to the previous sequence term to get the next term of the sequence.
And the constant value among the arithmetic sequence is known as the common difference of the sequence. This constant value is very helpful for making a sequence if the first term and the common difference are given.
For example, the sequence of whole numbers, odd numbers, even numbers, or natural numbers is in an arithmetic sequence. Because the distance among each number is constant in the series of whole numbers, odd numbers, even numbers, or natural numbers.
The sequence can be increasing or decrease depending upon the constant value. If the constant value of the sequence is positive then the sequence is increasing. While if the constant term of the sequence is negative then the sequence must be decreasing.
For example, if the starting value of the sequence is 2 and the constant difference is 3 then the sequence becomes of the form like:
2, 5, 8, 11, 14, 17, 20, 23, 26, ….
This sequence is known as the increasing sequence. While if the starting value is 31 and the common difference is -4 then the sequence is obtained like:
31, 27, 23, 19, 15, 11, 7, 3, …
This sequence is known as decreasing sequence.
Formulas of the arithmetic sequence
There are various formulas for an arithmetic sequence.
For finding the common difference or constant value, we subtract the leading number from the previous number and use an equation.
Common difference = d = xn – xn-1
(i) For finding the nth term of the arithmetic sequence, we use an equation.
Nth term = xn = x1 + (n – 1) d
(ii) For finding the sum of the arithmetic sequence, we use an equation.
Sum of the sequence = s = n/2 * (2x1 + (n – 1) d)
How to solve arithmetic sequence by using its formulas?
By using formulas, we can easily find the common difference, the nth term of the sequence, and the sum of the sequence. You can also use an arithmetic sequence calculator for the calculation of the nth term of the sequence and the sum of the sequence.
Example 1: For the nth term of the sequence
Find the 11th term of the given arithmetic sequence, 1, 7, 13, 19, 25, 31, 37, ….?
Solution
Step 1: Write the given arithmetic sequence.
1, 7, 13, 19, 25, 31, 37, ….
Step 2: Now select the first term and the common difference among them.
x1 = 1
d = 7 – 1
d = 6
Step 3: Take the general formula for finding the nth term of the sequence.
Nth term = xn = x1 + (n – 1) d
Step 4: Now put the given values in the above formula.
Nth term = xn = 1 + (n – 1)6
Nth term = xn = 1 + 6n – 6
Nth term = xn = 6n – 5
Step 5: We have to find 11th term, so put n = 11.
Nth term = x11 = 6(11) – 5
Nth term = x11 = 66 – 5
Nth term = x11 = 61
Example 2
Find the 112th term of the given arithmetic sequence, 7, 15, 23, 31, 39, 47, 55, ….?
Solution
Step 1: Write the given arithmetic sequence.
7, 15, 23, 31, 39, 47, 55, ….
Step 2: Now select the first term and the common difference among them.
x1 = 7
d = 15 – 7
d = 8
Step 3: Take the general formula for finding the nth term of the sequence.
Nth term = xn = x1 + (n – 1) d
Step 4: Now put the given values in the above formula.
Nth term = xn = 7 + (n – 1)8
Nth term = xn = 7 + 8n – 8
Nth term = xn = 8n – 1
Step 5: We have to find the 112th term, so put n = 112.
Nth term = x112 = 8(112) – 1
Nth term = x11 = 896 – 1
Nth term = x11 = 895
Example 3: For sum of the arithmetic sequence
Find the sum of the first 15 terms of the given arithmetic sequence, 3, 8, 13, 18, 23, 28, 33, 38, ….?
Solution
Step 1: Write the given arithmetic sequence.
3, 8, 13, 18, 23, 28, 33, 38, ….
Step 2: Now select the first term and the common difference among them.
x1 = 3
d = 8 – 3
d = 5
Step 3: Take the general formula for finding the sum of the given arithmetic sequence.
Sum of the sequence = s = n/2 * (2x1 + (n – 1) d)
Step 4: Now put the given values in the above formula.
Sum of the sequence = s = n/2 * (2(3) + (n – 1)5)
Sum of the sequence = s = n/2 * (6 + 5n – 5)
Sum of the sequence = s = n/2 * (1 + 5n)
Sum of the sequence = s = n/2 + 5n2/2
Step 5: We have to find the sum of the first 15 terms, so put n = 15.
Sum of the sequence = s = 15/2 + 5(15)2/2
Sum of the sequence = s = 15/2 + 5(15 x 15)/2
Sum of the sequence = s = 15/2 + 5(225)/2
Sum of the sequence = s = 15/2 + 1125/2
Sum of the sequence = s = 7.5 + 562.5
Sum of the sequence = s = 570
Summary
Now you are witnessed that arithmetic sequence is not a difficult topic. By following the above examples and formulas and using an online nth term calculator, you can easily solve any problem related to arithmetic sequence to find the common difference, the sum of the sequence, or the nth term of the sequence.
