Table of Contents
Arithmetic Progression (AP) is a sequence of numbers where the gap between any two consecutive terms stays the same. In simple terms, the numbers in an AP follow a regular pattern. Each number in the sequence is obtained by adding or subtracting a fixed amount from the previous number. For example, the sequence 2, 6, 10, 14, … is an arithmetic progression because each number is obtained by adding 4 to the previous one, maintaining a consistent difference of 4 between every two numbers. In real life, you can see APs in scenarios like an employee’s annual income, increasing by $5000 each year. This yearly income increase creates an arithmetic progression because the gap between consecutive annual incomes remains constant at $5000.
What is Arithmetic Progression?
An arithmetic progression, often referred to as an AP, is a sequence of numbers where each term is obtained by adding the same fixed number to the previous term. This fixed number is known as the common difference and is represented as ‘d’. The initial term of an arithmetic progression is typically denoted as ‘a’ or ‘a1’. For instance, consider the sequence: 1, 5, 9, 13, 17, 21, 25, 29, 33, and so on. This is an arithmetic progression because the difference between every two consecutive terms is the same, which is 4. In other words, when you subtract any term from the one before it, you always get 4 as the result (5 – 1 = 9 – 5 = 13 – 9 = and so on). Moreover, you can see that each term (except the first one) in this sequence is obtained by adding 4 to the previous term. In this particular arithmetic progression:
- The first term (a) is 1.
- The common difference (d) is 4.
In a broader sense, an arithmetic progression can be represented as: {a, a + d, a + 2d, a + 3d, and so on}. Applying this formula to our example, we have the following arithmetic progression: {1, 1 + 4, 1 + 2 × 4, 1 + 3 × 4, and so on}, which simplifies to {1, 5, 9, 13, and so on}.
Arithmetic Progression Formula
In an arithmetic progression (AP), the first term ‘a’ and the difference ‘d’ are crucial. You’ll often use these formulas to work with AP:
- Common Difference in AP:
The common difference ‘d’ between consecutive terms is found by subtracting one term from the next, like this: d = a2 – a1 = a3 – a2 = a4 – a3 = … = an – an-1
- Finding the nth Term in an AP:
To discover the nth term ‘an’ of an AP, you can use this formula: an = a + (n – 1)d
- Sum of n Terms in an AP:
Calculating the sum ‘Sn’ of the first ‘n’ terms is done with this formula: Sn = n/2 [2a + (n – 1)d] = n/2 (a + l) Here, ‘l’ is the last term in the arithmetic progression. These formulas are valuable tools for solving problems related to arithmetic progressions, helping you pinpoint specific terms or determine the sum of a range of terms in the sequence.
Arithmetic Progression Common Terms
An Arithmetic Progression (AP) is a sequence of numbers, like 6, 13, 20, 27, 34, … In an AP, we use two important terms:
- Initial Term: This is the first number in the sequence, represented as ‘a1’ or simply ‘a.’ For example, in the series 6, 13, 20, 27, 34, the initial term is 6, which we can write as ‘a1 = 6’ or ‘a = 6.’
- Common Difference: In an AP, each term after the first one is found by adding a fixed value to the preceding term. This fixed value is known as the ‘common difference,’ and we symbolize it with ‘d.’ So, if the first term is ‘a1,’ the second term is ‘a1 + d,’ the third term is ‘a1 + 2d,’ and so on.
For instance, in the sequence 6, 13, 20, 27, 34, each term (except the first) is obtained by adding 7 to the previous term, making the common difference ‘d = 7.’ Here are a few examples of APs along with their initial terms and common differences: 6, 13, 20, 27, 34, … has an initial term of 6 and a common difference of 7. 91, 81, 71, 61, 51, … has an initial term of 91 and a common difference of -10. π, 2π, 3π, 4π, 5π, … has an initial term of π and a common difference of π. -√3, -2√3, -3√3, -4√3, -5√3, … has an initial term of -√3 and a common difference of -√3
Arithmetic Progression Nth Term
In an arithmetic progression (AP), you can find the general term (or nth term) using this formula: an = a + (n – 1) d. Here’s how it works: Imagine you have a sequence like 6, 13, 20, 27, 34, and you want to find the nth term. Start by plugging in the first term (a1 = 6) and the common difference (d = 7) into the formula. That gives you an = 6 + (n – 1) * 7, which simplifies to 7n – 1. So, the general term for this AP is an = 7n – 1. Now, you might be curious why knowing this general term is useful. Let’s explore its practical applications.
Use of AP Formula for General Term
When you’re looking for a specific number in an arithmetic progression (AP), like the next number in a series, you can easily find it. For example, if you want to figure out the 6th number in the sequence 6, 13, 20, 27, 34, …, you just add 7 to the previous number (the common difference ‘d’). In this case, the 6th number is found by adding 7 to the 5th number, which is 34. So, the 6th number is 34 + 7, which is 41. But things get trickier when you need to find a term much further down the sequence, like the 102nd term. Doing this manually can be quite a hassle. To make it easier, you can use a formula for finding the nth term of an AP. Let’s say you want to find the 102nd term, and you know the initial term ‘a’ is 6, and the common difference ‘d’ is 7. You can use this formula: an = a + (n – 1) * d Plug in n = 102, a = 6, and d = 7: a102 = 6 + (102 – 1) * 7 = 6 + 101 * 7 = 713 So, the 102nd term in the sequence 6, 13, 20, 27, 34, …, is 713. This formula is handy because it allows you to find any term in the sequence without manually calculating all the preceding terms. Here’s a table showing different examples of arithmetic progressions (APs), their initial terms, common differences, and the corresponding general term for each case:
| AP Formula in General Term | |||
| Arithmetic Progression | First Term | Common Difference | General Term (nth term) |
| AP | a | d | an= a + (n-1)d |
| 91, 81, 71, 61, 51, . . . | 91 | -10 | -10n + 101 |
| π, 2π, 3π, 4π, 5π,… | π | π | πn |
| –√3, −2√3, −3√3, −4√3–,… | -√3 | -√3 | -√3 n |
These formulas make it easier to find any term in these sequences.
Arithmetic Progression Sum
Let’s take a closer look at an arithmetic progression (AP). An AP starts with an initial term, often denoted as ‘a1’ or just ‘a,’ and it follows a specific pattern where each term is obtained by adding a fixed amount called the common difference ‘d’ to the previous term. Now, if you want to find the sum of the first ‘n’ terms in an AP when you don’t know what the nth term is, you can use this formula: Sum of the first ‘n’ terms (Sn) = (n/2) * [2a + (n – 1)d] However, if you already know the value of the nth term (often represented as ‘an’), you can find the sum of the first ‘n’ terms with this formula: Sum of the first ‘n’ terms (Sn) = (n/2) * [a1 + an] These formulas are useful for working with arithmetic progressions and finding the sum of their terms.
