Introduction
In algebra, the arithmetic progression is used frequently for solving a given set of numbers. In this progression, there will be a difference between two consecutive terms, similar in the whole sequence and said to be the arithmetic progression.
The distance between two successive terms in a sequence is constant. Such as 1, 4, 7, 10, 13, 16, … is an arithmetic sequence in which the constant distance is 3 among each consecutive number. In this post, we’ll learn about the definition, formula, and examples for the arithmetic sequence.
Definition
A progression or a sequence that has a constant distance between the series of numbers is known as an arithmetic sequence of the given numbers. In mathematics, numbers that are written in an ordered list are known as a sequence.
Example:
- 2, 5, 8, 11, … is an AP with a common difference of 3.
- -5, -1, 3, 7, … is an AP with a common difference of 4.
For example, if the initial term of the sequence is 3 and the common distance between the terms is 6 then the sequence that is formed is 3, 9, 15, 21, 27, 33, 39, 45, 51, … this sequence is said to be the increasing sequence.
If the initial value of the sequence is 23 and the common difference among the term is -3 then the sequence that is formed is 23, 20, 17, 14, 11, 8, 5, 2, -1, … this sequence is said to be the decreasing sequence.
You can use an arithmetic sequence calculator to get the arithmetic sequences of the given initial value and common difference up to the nth term.
Arithmetic Progression Formula
The general formula for this kind of sequence is given below.
Arithmetic Progression for nth term = xn = x1 + (n – 1) d
- xn is the nth term of the sequence.
- n is the total number of terms.
- d id the common distance between consecutive numbers.
The common distance between the numbers can be taken by using a formula if it is not given.
The formula to calculate the common distance is:
Common distance = d = xn – xn-1
The sum of the sequence is taken by using a general formula.
Sequence’s sum = S = n/2 * (2x1 + (n – 1) d)
- xn is the nth term of the sequence.
- X1 is the initial term of the sequence.
- n is the total number of terms.
- d id the common distance between consecutive numbers.
Examples
Following are the solved examples of the arithmetic sequence.
Example 1: For the nth term of the sequence
Find the 13th term of 1, 8, 14, 20, 26, 32, 38, … by using the formulas of an arithmetic progression.
Solution
Step 1: First of all, write the given sequence.
1, 8, 14, 20, 26, 32, 38, …
Step 2: Take the initial term and the common difference of the given sequence.
Initial term = x1 = 1
Common distance = d = xn – xn-1
Common distance = d = 8 – 1
Common distance = d = 7
Step 3: Write the general equation of the arithmetic progression for calculating the nth term.
Arithmetic progression for nth term = xn = x1 + (n – 1) d
Step 4: Substitute the initial term and the common distance in the formula.
Arithmetic progression for nth term = xn = x1 + (n – 1) 7
Arithmetic progression for nth term = xn = 1 + (n – 1) 7
Arithmetic progression for nth term = xn = 1 + 7n – 7
Arithmetic progression for nth term = xn = 7n – 6
Step 5: Put n = 13 because we have to find the 13th term of the sequence.
Arithmetic sequence for nth term = xn = 7(13) – 6
Arithmetic sequence for nth term = xn = 91 – 6
Arithmetic sequence for nth term = xn = 85
Example 2: For the increasing sequence
Find the arithmetic sequence if the first term is 7 and the common distance is 4 by using the formulas of an arithmetic sequence.
Solution
Step 1: Write the initial term and the common difference of the given sequence.
Initial term = x1 = 7
Common distance = d = 4
Step 2: Write the general equation of the arithmetic sequence for calculating the nth term.
Arithmetic sequence for nth term = xn = x1 + (n – 1) d
Step 3: Substitute the initial term and the common distance in the formula.
Arithmetic sequence for nth term = xn = x1 + (n – 1) 4
Arithmetic sequence for nth term = xn = 7 + (n – 1) 4
Arithmetic sequence for nth term = xn = 7 + 4n – 4
Arithmetic sequence for nth term = xn = 4n + 3
Step 4: Put n = 1, 2, 3, 4, 5, 6, 7, …. to get the sequence of the terms.
For n = 1
x1 = 4(1) + 3
x1 = 4 + 3
x1 = 7
For n = 2
X2 = 4(2) + 3
X2 = 8 + 3
X2 = 11
For n = 3
X3 = 4(3) + 3
X3 = 12 + 3
X3 = 15
For n = 4
X4 = 4(4) + 3
X4 = 16 + 3
X4 = 19
For n = 5
X5 = 4(5) + 3
X5 = 20 + 3
X5 = 23
For n = 6
X6 = 4(6) + 3
X6 = 24 + 3
X6 = 27
For n = 7
X7 = 4(7) + 3
X7 = 28 + 3
X7 = 31
Step 5: Now write the calculated terms in the form of a sequence.
Arithmetic sequence = x1 + x2 + x3 + x4 + x5 + x6 + x7 + …
Arithmetic sequence = 7, 11, 15, 19, 23, 27, 31, …
You can also use an nth term calculator to get the sequence of the given initial term and the common distance. Follow the below steps to get the arithmetic sequence.
Step 1: Input the total terms, initial term, and the common distance.
Step 2: Hit the calculate button below the input box, the sequence will come in a couple of seconds.
Step 3: To view the step-by-step solution press the show more button.
Summary
In this article, we’ve discussed all the basics of the arithmetic sequence. Now you can grab all the basic concepts of the arithmetic sequence by learning the above post. You can easily solve any problem related to sequences.
What is an arithmetic progression in Class 10?
An arithmetic progression (AP) is a progression in which the difference between two consecutive terms is constant. Example: 2, 5, 8, 11, 14…. is an arithmetic progression.
How do you solve arithmetic progression?
- Common difference of an AP: d = an – an–1.
- nth term of an AP: an = a + (n – 1)d.
- Sum of n terms of an AP: Sn = n/2(2a+(n-1)d)
What is N in the arithmetic progression formula?
The sum of n terms of AP is the sum(addition) of the first n terms of the arithmetic sequence. It is equal to n divided by 2 times the sum of twice the first term – ‘a’ and the product of the difference between the second and first term-‘d’ also known as common difference, and (n-1), where n is numbers of terms to be added.
What is the arithmetic progression formula for the last term?
Thus nth term of an AP series is Tn = a + (n – 1) d, where Tn = nth term and a = first term. Here d = common difference = Tn – Tn-1. The sum of n terms is also equal to the formula where l is the last term.
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