The term or sequence arithmetic progression is the comparison between consecutive quantities that has a constant difference between them. For example, the sequences 5, 7, 9, 11, 13, 15, are considered an arithmetic progression with a difference of 2. Mathematical series or progressions are the sum or result of finite arithmetic progression and are maybe called arithmetic progressions or progressions. A positive value of “d” means the member terms grow towards positive infinity.
A negative value of “d” means the member terms grow towards negative infinity. The concept of progression is quite common in our daily lives, for example, counting the number of students in a class, the number of days in a week, or the number of months in a year. This series and sequence pattern has been categorized in Mathematics as progressions. Arithmetic Progression is a progression where every term can be described using a formula.
Different definitions for Arithmetic Progression:
- Mathematical sequences in which the difference between consecutive terms is always a constant. If the sequence is abbreviated AP, the differences are always constant.
- The arithmetic sequence or progression is defined as a series of numbers in which the second number can be obtained by adding one to the previous one for every pair of consecutive terms.
- As we have seen, the common difference of an arithmetic sequence is the number that must be added to each term to reach the subsequent term. Now, consider the sequence, 5, 8, 11, 14, 17, 20,…is considered as an arithmetic sequence with a common difference of 3.
Important elements of Arithmetic Progression:
- Common difference (d):
The term, ” d “, is obtained from the difference between two terms, for a given series of terms. Suppose, a1, a2, a3, …., and is an AP, then the difference between two terms ” d ” can be calculated as follows;
D = a2 – a1 = a3 – a2 = ……. = an – an – 1, here ‘d’ represents the common difference between two things. It can be positive, negative, or zero.
- The first term of an AP:
In addition to common differences, there are also AP’s that follow;
A, a + d, a + 2d, a + 3d, a + 4d, ………. ,a + (n – 1) d, This progression starts with a term called “a”.
Formulas for calculation of Arithmetic Progression:
During the learning process, we come across a couple of key formulas, and they are The nth term of Arithmetic Progression & Sum of the first n terms.
- The nth term of an AP:
To find the nth term of an AP, use the following formula:
an = a + (n – 1) × d, here d = Common difference, an = Nth term, a = First term & d = Common difference
Example:
Find the nth term of AP: 6,7,8,9,10,…., an, if the number of terms is 15.
Solution: Given, AP: 6,7,8,9,10…., an
n=15
By the formula we know, an = a+(n-1)d
First-term, a =6
The common difference, d=7-6=1
Therefore, an = 6+(15-1)1 = 6+14 = 20
- Sum of N terms of AP:
Any progression can be summarized with n terms. For an AP, the total number of terms can be summed with the first term and the total number of terms. The formula for the arithmetic progression sum can be found below.
S = n/2[2a + (n – 1) × d]
- Sum of AP when only first & last term is given:
To find the sum of AP when the first and last terms are given, use the following formula: S = n/2 (First term + Last term).
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