What is Arithmetic Progression?
A series of integers known as an arithmetic progression (AP) has a constant difference between each pair of terms. This common difference is known as the “common ratio.” For example, the sequence 2, 5, 8, 11, 14 is an arithmetic progression because the common difference between any two consecutive terms is 3.
Arithmetic progression has many applications in mathematics, science, and engineering. In mathematics, arithmetic progression is used in solving problems related to the sum of n terms, geometric progression, and infinite series. In science, arithmetic progression is used in calculating the time of a body in simple harmonic motion.
The formula S n = n/2 (2a + (n-1)d, where “a” stands for the first term, “d” stands for a common difference, and “n” stands for the total number of terms in the progression, calculates the sum of n terms in an arithmetic progression. This formula can be used to find the sum of an arithmetic progression without having to add all the terms individually.
What is Arithmetic Sequence?
An arithmetic sequence is a specific type of arithmetic progression where the difference between any two consecutive terms is a constant value. In other words, an arithmetic sequence is a progression where the common difference is a constant value. For instance, the arithmetic sequence 2, 4, 6, 8, and 10 is valid since the common difference is 2.
An arithmetic sequence is used in mathematical problem-solving and real-world applications. In mathematics, the arithmetic sequence is used to understand the concept of a linear function and linear equation. In real-world applications, the arithmetic sequence is used in financial mathematics to calculate an investment’s future value.
The arithmetic sequence also has a formula for the nth term, a_n = a_1 + (n-1)d , where a1 is the sequence’s first term, and n is the term’s position in the sequence, and d is a common difference. This formula can find any term of an arithmetic sequence without having to list out all of the terms individually.
Difference Between Arithmetic Progression and Arithmetic Sequence
- An arithmetic progression is a line of numbers in which the difference between any two immediate terms is a constant, whereas an arithmetic sequence is a specific kind of AP where the common difference is not zero.
- A general formula for the nth term of an arithmetic progression is an + d, where a denotes the first term and d signify a common difference, whereas the formula for the nth term of an arithmetic sequence is also an + d.
- An arithmetic progression can have any number of terms, whereas an arithmetic sequence must have at least two terms.
- The sum result of n terms of an arithmetic progression can be calculated using the formula: Sn = n/2 * (2a + (n-1)d), whereas the sum of an arithmetic sequence is calculated using the formula: Sn = n/2 * (a_1 + a_n)
- Arithmetic progression can have an infinite number of terms, whereas the number of terms in an arithmetic sequence can be finite or infinite.
Comparison Between Arithmetic Progression and Arithmetic Sequence
| Parameters of Comparison | Arithmetic Progression | Arithmetic Sequence |
| Definition | Sequence of Numbers | A Specific Kind of AP |
| General Formula | + d | Same as AP |
| Formula of Sum of n Terms | Sn = n/2 * (2a + (n-1)d) | Sn = n/2 * (a_1 + a_n) |
| Number of Terms | Any Number | At Least Two |
| Finite/Infinite | Infinite | Finite |