Arithmetic Progressions Formula: Properties, Common Term, Examples

Arithmetic Progressions Formula defined as An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms remains constant. In other words, the terms of an arithmetic progression follow a regular pattern where each term is obtained by adding (or subtracting) a fixed value to (or from) the previous term.

For instance, the sequence 2, 6, 10, 14, … is an arithmetic progression because each number in the sequence is obtained by adding 4 to the previous term, maintaining a consistent difference of 4 between each consecutive pair of terms.

In real-life scenarios, you can find examples of arithmetic progressions, such as the annual income of an employee who receives a fixed salary increase of $5000 each year. This increase in income each year forms an arithmetic progression because the difference between consecutive annual incomes remains the same ($5000).

What is Arithmetic Progression?

An arithmetic progression (AP) is a sequence of numbers where the differences between every two consecutive terms are the same. In this progression, each term, except the first term, is obtained by adding a fixed number to its previous term. This fixed number is known as the common difference and is denoted by ‘d’. The first term of an arithmetic progression is usually denoted by ‘a’ or ‘a1’.

For example, 1, 5, 9, 13, 17, 21, 25, 29, 33, … is an arithmetic progression as the differences between every two consecutive terms are the same (as 4). i.e., 5 – 1 = 9 – 5 = 13 – 9 = 17 – 13 = 21 – 17 = 25 – 21 = 29 – 25 = 33 – 29 = … = 4. We can also notice that every term (except the first term) of this AP is obtained by adding 4 to its previous term. In this arithmetic progression:

a = 1 (the first term)

d = 4 (the “common difference” between terms)

Thus, an arithmetic progression, in general, can be written as: {a, a + d, a + 2d, a + 3d, … }.

In the example mentioned earlier, we have the following arithmetic progression: {a, a + d, a + 2d, a + 3d, … } = {1, 1 + 4, 1 + 2 × 4, 1 + 3 × 4, … } = {1, 5, 9, 13, … }

Also Check – Data Handling Formula

Arithmetic Progression Formula (AP Formulas)

For the first term ‘a’ of an arithmetic progression (AP) and the common difference ‘d’, here are some commonly used AP formulas that are helpful for solving various problems related to AP:

Common Difference of an AP:

The common difference ‘d’ between any two consecutive terms of an AP is calculated as follows:

d = a2 – a1 = a3 – a2 = a4 – a3 = … = an – an-1

nth Term of an AP:

To find the nth term ‘an’ of an AP, you can use the formula:

a_n= a + (n – 1)d

Sum of n Terms of an AP:

The sum ‘Sn’ of the first ‘n’ terms of an AP can be computed using the formula:

S_n = n/2 [2a + (n – 1)d] = n/2 (a + l)

Here, ‘l’ represents the last term of the arithmetic progression.

These formulas are essential tools for solving problems involving arithmetic progressions, allowing you to determine specific terms or find the sum of a range of terms in the sequence.

The following image provides a visual representation of these AP formulas for easy reference.

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Also Check – Introduction to Euclid Formula

Common Terms Used in Arithmetic Progression

Starting now, we will use the abbreviation “AP” for arithmetic progression. An AP is typically represented as a sequence like a1, a2, a3, . . . and involves the following terminology:

Download PDF Arithmetic Progression Formula

Initial Term of Arithmetic Progression:

As its name implies, the initial term of an AP is the first number in the sequence. It is usually denoted as a1 (or simply a). For instance, in the series 6, 13, 20, 27, 34, . . . , the initial term is 6, which can be expressed as a1 = 6 (or) a = 6.

Common Difference of Arithmetic Progression:

In an AP, each term, except the first one, is generated by adding a constant value to the preceding term. This constant value is referred to as the “common difference” and is symbolized by ‘d.’ In other words, if the first term is a1, then the second term is a1 + d, the third term is a1 + d + d = a1 + 2d, and the fourth term is a1 + 2d + d = a1 + 3d, and so forth. For instance, in the sequence 6, 13, 20, 27, 34, . . . , each term, except the first one, is obtained by adding 7 to the preceding term. Therefore, the common difference in this case is d = 7. In general, the common difference represents the gap between any two consecutive terms in an AP. Consequently, the formula for calculating the common difference of an AP is: d = an – an-1.

Here are some examples of APs along with their initial term and common difference:

6, 13, 20, 27, 34, . . . is an AP with an initial term of 6 and a common difference of 7.

91, 81, 71, 61, 51, . . . is an AP with an initial term of 91 and a common difference of -10.

π, 2π, 3π, 4π, 5π, . . . is an AP with an initial term of π and a common difference of π.

-√3, −2√3, −3√3, −4√3, −5√3, . . . is an AP with an initial term of -√3 and a common difference of -√3.

Also Check – Polynomials Formula

Nth Term of Arithmetic Progression

The formula for the general term (or nth term) of an AP, where the first term is ‘a’ and the common difference is ‘d,’ can be expressed as follows: an = a + (n – 1) d. To illustrate, when we want to determine the general term (or nth term) for the sequence 6, 13, 20, 27, 34, . . . , we substitute the values of the first term, a1 = 6, and the common difference, d = 7, into the formula for the nth term. This results in the equation: an = a + (n – 1) d = 6 + (n – 1) * 7 = 6 + 7n – 7 = 7n – 1. Therefore, the general term (or nth term) of this AP is expressed as: an = 7n – 1.

Now, you might be wondering about the significance of determining the general term of an AP. Let’s explore its practical use.

Also Check – Introduction to Euclid Formula

Use of AP Formula for General Term

Certainly, when it comes to finding a specific term in an arithmetic progression (AP), we can simply add the common difference ‘d’ to the previous term. For instance, if we want to determine the 6th term in the sequence 6, 13, 20, 27, 34, . . . , we can easily find it by adding ‘d = 7’ to the 5th term, which is 34. Therefore, the 6th term is calculated as follows: 6th term = 5th term + 7 = 34 + 7 = 41.

However, the process becomes challenging when we need to find a term much further down the sequence, such as the 102nd term. Manual calculation can be quite cumbersome in such cases. To simplify this, we can utilize the formula for the nth term of an AP. For example, if we need to find the 102nd term and we have the initial term ‘a = 6’ and the common difference ‘d = 7,’ we can use the formula:

a_n = a + (n – 1) d

By substituting n = 102, a = 6, and d = 7 into the formula, we obtain:

a₁₀₂ = 6 + (102 – 1) * 7 = 6 + (101) * 7 = 713

Hence, the 102nd term of the given AP, 6, 13, 20, 27, 34, . . . , is 713. This highlights the importance of the general term (or nth term) formula in an AP, which serves as the explicit formula for the arithmetic sequence. It enables us to find any term in the sequence without the need to calculate its preceding terms manually.

Below is a table illustrating various examples of arithmetic progressions (APs), along with their initial term, common difference, and the corresponding general term for each case.

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

Also Check – Probability Formula

Sum of Arithmetic Progression

Let’s examine an arithmetic progression (AP) with an initial term a1 (also represented as ‘a’) and a common difference d.

When the nth term is not known, the sum of the first n terms of the AP can be calculated using the formula:

S_n= (n/2) * [2a + (n – 1)d]

However, when the nth term (an) is known, the sum of the first n terms can be found using the formula:

S_n = (n/2) * [a1 + an]

Example: Mr. Kevin has an annual income of $400,000, and his salary increases by $50,000 each year. What will be his total earnings at the end of the first 3 years?

Solution: Mr. Kevin’s earnings for the first year are represented as ‘a’ and equal to $400,000. The annual increment is ‘d,’ which amounts to $50,000. We need to calculate his total earnings over 3 years, so ‘n’ is set to 3.

By applying these values to the arithmetic progression (AP) sum formula:

S_n = n/2 [2a + (n – 1)d]

We can calculate as follows:

S_n = 3/2 [2(400,000) + (3 – 1)(50,000)]

= 3/2 [800,000 + 100,000]

= 3/2 [900,000]

= $1,350,000

Therefore, Mr. Kevin will earn $1,350,000 in the first 3 years.

Alternatively, we can arrive at the same result through a simpler calculation: Mr. Kevin’s earnings in 3 years would be $400,000 + $450,000 + $500,000 = $1,350,000. This manual calculation is feasible when ‘n’ is a smaller value. However, the above formulas become particularly useful when ‘n’ is a larger value.

Important Notes on Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers where each term is generated by adding a constant value to the preceding term. In this context, ‘a’ represents the first term, ‘d’ signifies the common difference, ‘an’ stands for the nth term, and ‘n’ indicates the total number of terms.

• In a general representation, an AP can be expressed as follows: a, a + d, a + 2d, a + 3d, …
• The nth term of an arithmetic progression can be determined using the formula: an = a + (n − 1)d.
• To find the sum of an AP, you can use the formula: S_n= n/2 [2a + (n − 1)d].
• Graphically, the representation of an AP results in a straight line, with the slope of this line being equivalent to the common difference.
• It’s worth noting that the common difference in an AP doesn’t necessarily have to be positive. For instance, consider the sequence 16, 8, 0, −8, −16, … In this progression, the common difference is negative, as it can be calculated as follows: d = 8 – 16 = 0 – 8 = -8 – 0 = -16 – (-8) =… = -8.

Examples Related To AP

Example 1: Determine the general term for the arithmetic progression -3, -(1/2), 2…

Solution:

Given the sequence: -3, -(1/2), 2…

Here, the first term is represented as ‘a’ and is equal to -3, while the common difference ‘d’ can be calculated as follows: d = -(1/2) – (-3) = -(1/2) + 3 = 5/2

Using the formula for the general term of an arithmetic progression:

a_n= a + (n – 1) * d

a_n = -3 + (n – 1) * (5/2)

= -3 + (5/2)n – 5/2

= (5/2)n – 11/2

Hence, the general term for the given arithmetic progression is:

 Answer: a_n = (5/2)n – 11/2

Example 2: Determine which term of the arithmetic progression 3, 8, 13, 18,… is equal to 78.

Solution:

Given the sequence: 3, 8, 13, 18,…

Here, the first term ‘a’ is 3, and the common difference ‘d’ can be calculated as follows: d = 8 – 3 = 5

Let’s assume the term we’re looking for is represented by ‘an’ and is equal to 78.

Using the formula for the general term of an arithmetic progression:

a_n = a + (n – 1) * d

Substitute the known values:

78 = 3 + (n – 1) * 5

78 = 3 + 5n – 5

78 = 5n – 2

Now, isolate ‘n’:

5n = 78 + 2

5n = 80

n = 16

Therefore, the term 78 corresponds to the 16th term in the given arithmetic progression.

Answer: ∴ 78 is the 16th term.

Example 3: Find the sum of the first 5 terms of the arithmetic progression with a first term of 3 and a fifth term of 11.

Solution: Given a1 = a = 3, a5 = 11, and n = 5.

Using the formula for the sum of the first n terms of an arithmetic progression (Sn):

S_n = (n/2) * (a + an)

Substitute the known values:

S5 = (5/2) * (3 + 11)

= (5/2) * 14

= 35

Answer: The sum of the first 5 terms of the arithmetic progression is 35.