Arithmetic progression is a sequence of numbers that follow a pattern. For example, 1, 3, 5, 7, 9.
The pattern in this arithmetic progression is that each number is the sum of the two numbers that come before it.
Arithmetic Progression Definition
Arithmetic progression is a sequence of numbers in which the difference between any two consecutive members of the sequence is constant.
This is an example of arithmetic progression:
2, 4, 6, 8, 10
The common difference between any two consecutive members of the sequence is 2.
Arithmetic progression is a sequence of values in which the difference between any two consecutive members is constant.
The simplest form of an arithmetic progression is a sequence of natural numbers, such as 3, 5, 7, and 9.
An arithmetic progression can be finite or infinite.
The first five terms in an arithmetic progression are a finite sequence if there are only five terms.
An infinite sequence of numbers has no end point and goes on forever in both directions.
Arithmetic Progression Formula
The arithmetic progression formula is used to find the sum of a series of consecutive integers.
The formula is as follows:
The first term, “a” is the first number in the series. The last term, “n”, is the final number in the series. The common difference between consecutive terms is “d”. The sum of all numbers in an arithmetic progression can be calculated by using this formula.
The arithmetic progression formula can be written as:
A = P + (A1-P)
Common Terms Used in Arithmetic Progression
The common terms in arithmetic progression are:
- first term: the first number in the sequence
- second term: the second number in the sequence
- common difference: the difference between consecutive terms.
Important Notes on Arithmetic Progression
· a is represented as the first term, d is a common difference, and is the nth term, and n is the number of terms.
- the nth term of an AP can be obtained as an = a + (n−1)d
- An AP is a list of numbers in which each term is obtained by adding a fixed number to the preceding number.
- The common difference doesn’t need to be positive always. For example, in the sequence, 16,8,0,−8,−16,…. the common difference is negative (d = 8–16 = 0–8 = -8–0 = -16 — (-8) =… = -8).
Arithmetic Progression Examples
Arithmetic progression examples are used to solve problems involving arithmetic sequences.
In the following examples, we will see how to find the sum of an arithmetic sequence with a given starting number and a given interval.
Example 1: 2, 5, 8, 11, 14
We will find the sum of this sequence by adding up all the numbers from 2–14. The sum is 55.
The first term is 2 and the last term is 14 so there are 12 terms in total. The sum of any arithmetic progression with an even number of terms can be found by adding the first and last terms together and then multiplying that result by two. In this case, we have (2+14)*2=55 This rule also applies when there is an odd number.
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