Sequences appear in all forms of everyday life. Some simple examples include the time on a clock or the number of reps steadily increasing in your exercise routine. It is therefore important that you get to grips with this mathematical concept as soon as possible. In this guide, we explain everything you need to know about arithmetic progression, from what it is to how to use the explicit formula to calculate the Nth term.
What is arithmetic progression?
Arithmetic progression (AP) is an arithmetic sequence in which every number has a common difference. This is one of the simplest types of sequences within the maths world, as each consecutive term is increasing by the same amount.
Here are a few examples of arithmetic progressions:
2, 4, 6, 8, 10, 12, 14, ….
3, 8, 13, 18, 23, 28 ….
Let’s look at the two examples above. They can both be referred to as successive terms within an arithmetic progression because each number in the arithmetic series increases by the same number.
For the first example, the common difference is 2.
Because, 2 + 2 = 4 then 4 + 2 = 6 then 6 + 2 = 8 and so on
Similarly, with the second example, there is also a common difference, but this time, the consecutive terms are added by 5 each time.
So, 3 + 5 = 8 then 8 + 5 =13 then 13 + 5 = 18 and so on.
Properties of arithmetic progressions
Just like with every element of maths, AP comes with its own set of properties that are specific to this concept and allows you to recognize when a sequence is an arithmetic progression, and not an alternative type of sequence, such as a geometrical progression, for example.
Here are the key properties linked to AP:
- If the same number is added from the previous term of an AP, the resulting terms in the sequence are also in AP using the same common difference.
- If the same number is subtracted from the previous term of an AP, the resulting terms in the sequence are also in AP using the same common difference.
- If each number in an AP is divided with the same non-zero number, then the resulting sequence is also in an AP.
- If each number in an AP is multiplied with the same non-zero number, then the resulting sequence is also in an AP.
- Three numbers x, y and z are in an AP if 2y = x + z
- A sequence is an AP if its nth term is a linear expression.
- If we select terms in the regular interval from an AP, these selected terms will also be in AP.
Arithmetic progression formulas
Once you are able to decipher the common difference amongst a set of numbers within an AP sequence, understanding the formula becomes easier. There are actually a few different types of formulas that you can use when looking at arithmetic progression.
The general form of AP
First off, we have the general form of AP, which is the most basic formula related to arithmetic progression. It simply lays out, in mathematical terms, the process occurring in the sequence.
The formula is as follows:
a, a + d, a + 2d, a + 3d, . . .
In this formula, a is the initial term and d is the common difference.
So, working from the initial term, you have that number plus the difference. Then, to calculate the third number in the sequence, you have to take the initial number and multiply the difference by 2 and add it to your initial term… and so on and so forth.
The Nth term of AP
This formula is slightly more complicated, but it can be used in order to find any number in the sequence, making it an extremely useful formula to understand.
The formula is as follows:
an = a + (n – 1) × d
Although this might look complicated, for any problem in which you need to decipher the Nth term, you will be given all the information you need. All you have to do is input it into the formula.
Let’s look at an example:
Find the value of n, if a = 10, d = 5, an = 95.
an = a + (n − 1) × d
95 = 10 + (n − 1) × 5
(n − 1) × 5 = 95 – 10 = 85
(n − 1) = 85/ 5
(n − 1) = 17
n = 17 + 1
n = 18
Sum of arithmetic progression
Increasing in complexity, this final formula is the most confusing of them all, but don’t let the onslaught of letters and numbers confuse you, because it is actually just as simple as the previous formula. All you need to do is substitute the numbers into the formula.
The formula is as follows:
S = n/2[2a + (n − 1) × d]
a = initial element
d = common difference
n = N-th term
Class solution problems are usually built into a wordy problem, where you need to pick out the relevant information and insert it into the formula. Here is a quick example:
Mr. Smith earns $400,000 per year. He negotiated his salary to earn an extra $50,000 per year. How much does he earn at the end of the first 3 years?
The only important information in this question is the numbers. You can therefore largely ignore all the contextual words surrounding the numbers. All you need to know is that he currently makes $400,000 per year, making this his initial term. His wage increases by $50,000 per year, which will be the common difference, and you have to calculate his salary after 3 years, making 3 the number in the sequence you are adding up to.
When inputting it into the formula, it looks as follows:
Sn= 3/2(2(400000)+(3-1)(50000))
= 3/2 (800000+100000)
= 3/2 (900000)
= 1350000
We therefore know that he made $1,350,000 in the three years he’s been at the company.
Common terms used in arithmetic progression
There are some common terms that are frequently used when discussing AP. In order to obtain a comprehensive understanding of arithmetic progression, you first need to understand what all these words and phrases relate to.
First term
This refers to the first number in the sequence. This first number will always be your starting point and is essential in virtually every AP formula.
Common difference
A common difference is the main thing that differentiates an arithmetic progression from other sequences such as a geometrical progression. All it is referring to is the fact that each of the consecutive terms in the sequence increases by the same number.
Nth term
When receiving questions about arithmetic progression, more often than not, you will be asked to find the Nth term of a sequence. This means that you are being asked to find a specific number within the sequence. The N is a substitute for the number you are being asked to find. For example, you might need to find the 10th, 100th, or 1000th term. N is the substitute in the formula that you use to calculate these numbers.
Differences between arithmetic progression and geometric progression
People often get confused between all the different mathematical sequences, and wrongly assume they have the same properties. However, this assumption will lead you to make mathematical errors as you mix up formulas and numerical elements.
One of the most common sequences that get mixed up is arithmetic progression and geometric progression. In order to clear up any confusion, we have collated a list of all the key differences between the two.
- GP has a common ratio, AP doesn’t
- AP has a common difference, GP doesn’t
- The next in the sequence is the product of the previous number and common ratio
- The next in the sequence is the sum of the previous number and the common difference
- The GP variation is non-linear
- The AP variation is linear
Math tutoring services
It is impossible to escape arithmetic progression, as it is sure to show up in your mathematical tests in school. It also has an abundance of real-life applications, which means it is important that you understand the basics surrounding this mathematical concept. The tutors at Tutorax are dedicated to making sure that you reach your full academic potential. We offer both in-person and online tutoring so that your sessions can seamlessly fit into your schedule.