Arithmetic Progression (AP) refers to a sequence or series of numbers wherein the difference between every two consecutive numbers is the same. For example, consider the following series: 1, 2, 3, 4, 5, 6. It follows a pattern with 1 as a common difference.
This concept is applied in many aspects of daily life, such as to calculate taxi fares, employee income, stadium seats, and so on and so forth. Concepts like these should be taught in schools in an easy and comprehensible manner. In this article, we will walk you through the AP formulas and terms used and explain how to find out the AP of a sequence.
Common Terms used in an AP
The common terms used in AP are:
- first term: a
- common difference: d
- nth term: an
- sum of n terms: Sn
First Term of Arithmetic Progression
Consider an AP sequence: a1 , a2, a3,… an. Here the first term of the series a1 is taken as ‘a’. The sequence can also be written in terms of common difference as:
| a, a+d, a+2d, a+3d,….., a+(n-1)d |
Example: Consider the sequence 22, 44, 66, 88, 110.
Here the first term a = 22.
Common Difference in Arithmetic Progression
The common difference in the Arithmetic Progression refers to the difference between two successive terms. To obtain the next term in the series, this common difference is added to the preceding term. It is represented by ‘d’, which can be positive, negative or zero.
In the AP sequence a1, a2, a3,…, an – 1, an.
| d = a2 – a1 = a3 – a2 = an – an – 1. |
Example: Let us take the example of 1, 3, 5, 7, 9, 11.
Common difference d = 3 – 1 = 2 or 11 – 9 = 2.
nth Term of Arithmetic Progression
The last term in the Arithmetic Progression is called the nth term. To find the nth term an,
| an = n + (n-1)*d |
‘n’ represents the number of terms in the series.
For example, to find the nth term in the series 3, 6, 9, 12, 15,… an with 10 terms, we use the above formula. On comparison we get,
a (first term) = 3
d (common difference) = 6 – 3 = 3.
n (number of terms) = 10
By the formula, an = 10 + (10-1)*3 = 37.
Sum of Arithmetic Progression
The sum of n terms can be calculated if the first term, the common difference and the number of terms in the sequence are given:
| Sn = n/2 [2a + (n-1)*d] |
Example: Let us consider the series 5, 10, 15, 20, 25, 30.
Here,
a = 5
d = 10 – 5 = 5
n = 6
We know that Sn = n/2 [2a + (n-1) *d].
Substituting the values, we get
Sn = 6/2 [2*5 + (6 – 1) * 5]
= 3 [10 + 5 * 5]
= 3 [10+25]
= 3 [35]
= 105
If the first term and last term are given, then we have the formula:
| Sn = n/2 (first term + last term) |
Considering the same example,
First term = 5
Last term = 30
Then, Sn = 6/2 (5+30).
= 3 (35)
= 105.
Derivation of AP Sum Formula
To derive the sum of the AP formula, consider an AP sequence a, a+d, a+2d, a+3d,… a+(n-1)*d consisting of ‘n’ terms. Assume that Sn is the sum of these numbers. It can be found out in two ways:
(i)Sn = a + (a + d) + (a + 2d) +… + [a + (n – 1) × d] —————— (i)
Or writing the terms in reverse order, we have:
(ii)Sn = [a + (n-1) × d] + [a + (n-2) × d] + [a + (n-3) × d] + ……. (a) ———– (ii)
Adding both the equations term-wise, we have:
2Sn = [2a + (n – 1) × d] + [2a + (n – 1) × d] + [2a + (n – 1) × d] +…………. + [2a + (n – 1) ×d] (n terms)
2Sn = n × [2a + (n – 1) × d]
Sn = n/2 [2a + (n – 1) × d]
Arithemtic Progression Formulas (AP)
Since we are well-versed with the terms used in the AP sequence and have covered all the formulas along with the meaning of the terms in detail, it becomes easy to understand and remember the formulas. Let us recall these formulas:
| Common difference formula (d) = a2 – a1 = an – an – 1Formula to find nth term (an) = a + (n-1)*dFormula to calculate the sum of n terms (Sn) = n/2 [2a+(n-1)d] |
Use of AP Formula for General Term
If the sequence is small, then it is easy to find the next term just by adding the common difference to the previous term. But what if you want to find the 100th or 200th term in an infinity series? Doesn’t it get difficult? In that case, we use the an = a + (n-1) * d formula, substituting n as 100 or 200, respectively.
For example, consider this series: 7, 14, 21, 28,…
If you want to find out the 100th term then,
a100 = 7 + (100 – 1) * 7
= 7 + (99) * 7
= 7 + 693
= 700.
Therefore, the general term formula is used to calculate any term in the given series, even if the previous term is unknown.
Let us look at two more examples:
| Arithmetic Progression | First Term | Common Difference | General Term |
| 62, 64, 66, 68 | 62 | 2 | a8 = 62 + (8-1)*2 |
| 600, 500, 400, 300, 100, | 600 | -100 | a25 = 600 + (25-1)*(-100) |
Arithmetic Progression Examples
Let us look at some examples with detailed solutions to obtain concept clarity.
Example 1: Find out the 99th term in the sequence 3, 5, 7, 9,…
Solution: To find out an, we have an = a+(n-1)d.
Here an = a99 , a = 3, d = 5-3 = 2, n = 99.
a99 = 3 + (99-1)*2
= 3 + (98*2)
= 3 + 196
= 199.
Example 2: Find the value of n if a = 100, d = 50 and an = 500.
Solution: Given a = 100, d = 50, and an = 24.
We have an = a + (n-1)*d.
Substituting the values, we get
500 = 100 + (n-1)* 50
500 = 100 + 50n – 50
500 – 100 + 50 = 50n
450 = 50n
n= 450/50
n= 9.
Example 3: Find the sum of the first 10 multiples of 10.
Solution: The first 10 multiples of 10 are 10, 20, 30, 40, 50, 60, 70, 80, 90, 100.
We know Sn = n/2 [2a + (n-1)*d].
On comparison, we get a = 10.
d = 10.
n = 10.
S10 = 10/2 [(2*10) + (10-1)*10]
S10 = 5 [20 + 90]
S10 = 5*110
S10 = 550.
Practice Questions on Arithmetic Progression
It is now time to pick your brains and test your knowledge on AP. Solve these practice problems to get a hold of the concept.
Question 1: Find the common difference of the progression 35.5, 33.5, 31.5, 29.5, 28.5.
Question 2: Find the 501st term if a = 1, d = – 13.
Question 3: Find the sum of the first 8 odd numbers.
Conclusion
Arithmetic progression is an important math concept that is easy and has only limited formulas. This basic concept is applied to find solutions for advanced and complex math problems as well. Practice a larger number of questions in group studies and revise the formulas regularly to get a hang of the topic.
Frequently Asked Questions (FAQs)
What Are the Differences Between Arithmetic Progression and Geometric Progression?
In the arithmetic progression, the difference between two successive terms is the same. Linear progression is observed in this pattern. Whereas in geometric progression, the ratio of two successive terms is the same. This follows exponential growth. The formula to calculate the next term and sum also differs.
How to Find the Sum of Arithmetic Progression?
If the first term (a), the common difference (d), and the number of terms (n) in the sequence are given, the formula to find the sum is
Sn = n/2 [2a + (n-1)*d]
What are the Types of Progressions in Maths?
In math, there are three important progressions, namely—arithmetic (AP), geometric (GP) and harmonic progression (HP).
What is Infinite Arithmetic Progression?
An infinite arithmetic progression is a series that continues infinitely. Usually the last term is unknown.
