A sequence is just a listing of numbers in which each term is created by adding a fixed term to each of its preceding numbers, except for the initial one, and the sequence continues indefinitely. To find an arithmetic series, use the formula for the nth term, using the initial terms, numbers of terms, and the common difference of the arithmetic sequence. In this article, we will discuss how to find the common difference of the arithmetic sequence using the first terms and using the sum formula. So, let’s begin.
1. What is the Common Difference and Sequence?
One way to define the common difference and sequence is by using a series as follows:
1,2,3,4,5…..n
- where the first term of the arithmetic sequence is a1 and
- the difference between the two consecutive terms is specified as the letter d.
This is known as the common difference of the arithmetic sequence. (See What is a Mathematical Sentence Example?)
2. What is the Formula of Common Difference of the Arithmetic Sequence?
The value that distinguishes one successive number from the next in an arithmetic series is a common difference. The formula to get the common difference of the arithmetic sequnce is: d = a(n) − a(n − 1)
- where a(n) is the nth term in the sequence and
- a(n − 1) is the preceding term or (n − 1)th term in the sequence. (See What is 20% of 70?)
3. How to find the Common Difference of the Arithmetic Sequence?
Photo by Austris Augusts on UnsplashCalculate the difference between the first and second terms of the AP by using the following subtraction formula:
d = a_{2} − a_{1}
- where d denotes the most common deviation.
- a_{2} is synonymous with any term that is not the first term.
- a_{1} refers to the previous term.
Let us consider example 15,21,27…..
a_{1} is the first term, which is 15, the value of a_{2} is 21, while a_{3 }is 27.
In this arithmetic series, the symbol d represents the common difference, which is 6, because
27 − 21 = 6 and 21 − 15 = 6, so both equals 6. (See What are the Types of Functions?)
4. How can you find the Common Difference in Arithmetic Sequence with Two Terms?
Let’s assume the second and sixth terms of an arithmetic sequence or progression are 2 and 30 respectively, i.e
a_{2 }= 2
a_{6 }= 30.
a_{2} value is already given, and if you add a common difference, you can obtain a_{3 }i.e. a_{3} = a_{2 }+ d.
The same process may be used to determine the remaining terms.
a_{4} = a_{3 }+ d
a_{5} = a_{4 }+ d
and similarly, a_{6} = a_{5} + d
Therefore, from a to a_{6}, additional 4 ds are appended. So, you can also use this direct formula to get the sixth term,
a_{6} = a_{2 }+ 4 d
In this way, you can find the common difference of the arithmetic sequence using the 2 terms. Also, check out What does 2/3 plus 2/3 equal?
5. How can you find First Term of Arithmetic Sequence?
Considering the order of 3, 7, 9,…. the number 3 comes first in this progression of terms. To simplify the definition of the nth term, we’ll just refer to the first term as a. The difference of number terms, which is denoted by the letter d, appears to be the most prevalent variant across this series.
If the initial term is to be determined in an arithmetic series using the sum formula, then use,
a = T_{n} − (n − 1) d
where T_{n }is the nth term, a is the first term of arithmetic sequence, d is the common difference of the arithmetic sequence and n is the number of terms. (See What is the Very Last Number in the World?)
6. How can You calculate Arithmetic Sequences?
Photo by Jessica Lewis on UnsplashIn most cases, the formula for an arithmetic series looks like this: a(n) = a_{1 + }(n − 1)d. An arithmetic sequence is a real-life concept that refers to a listing of numbers, in which the difference is always constant which helps to find the sum and the last term of the sequence. Must read What are Some Examples of Categorical Variables?
7. How can you find the Common Difference in Arithmetic Sequence Sum and Term?
Consider the following scenario,
a_{11 }= 30 (value of the 11th term)
S_{11 }= 55 (sum of first 11 terms)
n = 11
d =?
To find the common difference, first, apply the Arithmetic Progression formula, a(n) = a_{1 +} (n − 1)d
- a_{11 }= a_{1 }(11 − 1) d
- 30 = a_{1 }+ 10d, lets take it as Equation (1)
The second step is to utilize the formula for the sum of the arithmetic series.
- S = n/2 [2a + (n − 1) × d]
- S_{11} = 11/2 [2 a_{1}+ (11 − 1) × d]
- 55 = 11/2 [2a_{1} + 10d]
- 55 = 11/2 [2(a_{1} + 5d)]
- 55 = 11 (a_{1} + 5d)
- 5 = a_{1} + 5d, Equation (2)
Finding solutions by subtracting equation 2 from 1, we have,
- 30 = a_{1}+10d
- 5 = a_{1} + 5d,
- 25 = 5d
- d = 5, is the required common difference.
8. How can You find the Common Difference of an Arithmetic Sequence given the Sum?
Photo by Mika Baumeister on UnsplashWhen the sum is given, the common difference of an AP can be calculated by using
S = n/2 [2a + (n − 1) × d]
9. What is the Common Difference between AP 127135143151?
The first term in the AP problem, which is given as 127, 135, 143, 151, is 127, which is a_{1}. Similarly, a_{2 }= 135, The common difference can be determined by applying the formula d = a_{2} − a_{1} = 135 − 127 = 8. (Also read Why is Mathematical Concept Important?)