An Introduction to arithmetic progression: Definition, formulas, Applications, and Calculations:
In mathematics, arithmetic progression is a fundamental concept that is frequently used in algebra and number theory to deal with the study of relationships and patterns of numbers. It is also used in finance, physics, computer science, and gaming. Also Check: How to Write an Academic CV or Resume For Scholarship to Study Abroad
In this article, we’ll discuss the basics of the arithmetic sequence such as definition, formulas, applications, and solved examples.
What is arithmetic progression?
Arithmetic progression is a fundamental concept in mathematics that deals with a sequence of numbers in which the next term of the sequence is determined by taking the sum of the previous term and common difference.
For example,
2, 5, 8, 11, 14, 17, 20, …
2 is the first term of the sequence and 3 is the common difference of the sequence.
Abbreviations of an arithmetic sequence
| The 1st term | b1 |
| The common difference | d |
| The nth term | bn |
| The sum of the first ‘n’ terms | S(n) |
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Formulas for Arithmetic Progression
There are various important formulas for arithmetic progression. Some of the most common formulas are:
| Formula for | Formula |
| The formula for the nth term | bn = b1 + (n – 1) d
|
| The formula for the sum of the first ‘n’ terms | S(n) = n/2[2b1 + (n – 1) d]
|
| The formula for the sum of an infinite number of terms: | S(infinity) = b1/(1 – d) |
| The formula for common difference | d = bn – bn-1 |
Applications of Arithmetic Progression
Arithmetic sequences have various applications in different fields of mathematics, science, and engineering. Some of the most common applications of arithmetic progressions are:
| Application in | Uses |
| Financial Mathematics | In finance, the arithmetic progression is frequently used to model the growth of investments and stocks.
For assistance, an investor can get help from arithmetic progression to calculate the growth of their savings over time. |
| Algebra | In algebra, the arithmetic progression is widely used to evaluate different equations like linear, quadratic, and complex equations.
For assistance, the arithmetic progression formula for the nth term is helpful for solving linear equations problems. |
| Physics | In physics, arithmetic progression is a way to model the motion of objects with constant acceleration.
For instance, if an object moves with a constant acceleration ‘a’, then its velocity can be modeled by an arithmetic progression with the common difference d. |
| Computer Science | Arithmetic progression is used in computer science to optimize algorithms, especially in sorting and searching algorithms.
For instance, the binary (0, 1) search algorithm uses an arithmetic progression to search for an element in a sorted array. |
| Statistics | In statistics, arithmetic progression is used to represent various types of data, such as time series data and financial data.
For instance, an arithmetic progression can be used to model the growth of revenue of a company over time. |
| Engineering | Arithmetic progression is used in engineering to model various physical systems, such as electrical circuits and mechanical systems. For instance, an arithmetic progression can be used to model the flow of current through an electrical circuit. |
Uses of arithmetic progression
Arithmetic progression is an essential mathematical concept that has many practical applications. We use arithmetic progression because they help us to:
| Use to | Uses |
| Identify patterns | Arithmetic progression helps us identify patterns in numbers, making it easier to understand and analyze numerical data. |
| Predict outcomes | Arithmetic progression allows us to calculate future outcomes based on the data we have already collected.
For instance, we can use an arithmetic progression to guess the growth of a population over time. |
| Solve problems | Arithmetic progression provides us with formulas and techniques that allow us to solve a wide range of problems, from basic math equations to more complex mathematical and scientific calculations. |
| Simplify calculations | Arithmetic progression helps us simplify calculations by providing us with a systematic approach to dealing with numerical data. This makes it easier to perform calculations accurately and efficiently. |
| Model real-world situations | Arithmetic progression allows us to model real-world situations and solve problems related to these situations.
For instance, we can use an arithmetic progression to model the growth of a business. |
How to calculate arithmetic progression problems?
The problems of arithmetic progression involve the calculations of finding the nth term and sum of the n terms.
For finding the nth term
Example 1
Find the 44th term of 3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, … with the help of arithmetic progression for the nth term.
Solution
Step 1: First of all, write the given sequence and take the first term of the sequence and evaluate the common difference of the sequence.
Given sequence = 3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, …
b1 = 3
d = b2 – b1
d = 7 – 3
d = 4
Step 2: Now write the formula for finding the nth term of the sequence.
nth term of the sequence= bn = b1 + (n – 1) * d
Step 3: Now put the n = 44, the first term of the sequence “b1”, and common difference “d” to the above formula.
44th term of the sequence = b44 = 3 + (44 – 1) * 4
44th term of the sequence = b44 = 3 + (43) * 4
44th term of the sequence = b44 = 3 + 172
44th term of the sequence = b44 = 175
The nth term of any sequence can also be evaluated with the help of an online nth-term calculator by AllMath (https://www.allmath.com/arithmetic-progression.php).
Example 2
Find the 104th term of 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, … with the help of arithmetic progression for the nth term.
Solution
Step 1: Write the given comma-separated sequence, take the first term “b1” of the sequence, and determine the common difference “d” of the sequence.
Given sequence = 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, …
b1 = 21
d = b2 – b1
d = 27 – 21
d = 6
Step 2: Now take the formula.
nth term of the sequence= bn = b1 + (n – 1) * d
Step 3: Now put the n = 104, the first term of the sequence “b1”, and common difference “d” to the above formula.
104th term of the sequence = b104 = 21 + (104 – 1) * 6
104th term of the sequence = b104 = 21 + (103) * 6
104th term of the sequence = b104 = 21 + 618
104th term of the sequence = b104 = 639
The nth-term calculation of a sequence can be solved with the help of an arithmetic sequence calculator.
For finding the sum of the sequence
Example 1:
Find the sum of the first 50 terms of the 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, …
Solution
Step 1: Write the given comma-separated sequence, take the first term “b1” of the sequence, and determine the common difference “d” of the sequence.
Given sequence = 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, …
b1 = 32
d = b2 – b1
d = 36 – 32
d = 4
Step 2: Now take the formula.
Some of the sequence = s = n/2 * (2b1 + (n – 1) * d)
Step 3: Substitute the 1st term “b1 = 32”, common difference “d = 4”, and the total number of n values to the above formula of the finding Sum of the sequence.
Some of the first 50 terms = s = 50/2 * (2(32) + (50 – 1) * 4)
= s = 50/2 * (2(32) + (49) * 4)
= s = 50/2 * (64 + 49 * 4)
= s = 50/2 * (64 + 196)
= s = 50/2 * 260
= s = 25 * 260
= s = 6500
Wrap Up
The arithmetic sequence is a well-known sequence in mathematics that has a wide range of uses and applications. The formulas for finding the nth term and sum of the sequence are helpful for solving the problems of an arithmetic sequence.
