Arithmetic Progressions

Arithmetic Progressions are basically series that fall in sequences. The sequences are separated by common differences. In other words, it can be termed that the previous numbers in the sequences can be obtained by adding the common differences. There are certain formulas that can be used to solve the problems concerning the Arithmetic Progression series. There is also a possible formula from which the last or the nth term can be obtained.

Arithmetic Progressions

The progression is a series or sequence where the arrangement of terms is in such order that there is a definite relation between the consecutive terms. There are three types of progressions in mathematics such as Arithmetic progressions or AP, Geometric progressions or GP, and Harmonic progressions or HP. The Arithmetic Progressions are different from the GP or the HP series. The Arithmetic Progressions are used in daily lives to find the number of stunts in a respective class, the number of months in a year, or even the number of days in a week. The Arithmetic Progressions has terminologies such as common difference, first term, last term, and sum of n elements which is denoted by d, a, an, Sn respectively.

The general form of Arithmetic Progressions is [a, a+d, a+2d… a+ (n-1)d].

First-term is generally denoted as “a” in the Arithmetic progression. For example, in a series 6, 13,20,27,34, the first term is 6. The common difference in the given series will be 13-6, which is equal to 7. The common difference or d is fixed for every consecutive number in the series. The formula for finding the common difference d is an- a (n-1) where the difference can be obtained from the addition of the previous term. In a given Arithmetic progression, series if the value of common differences is positive then the Arithmetic Progression series will definitely be positive. In an Arithmetic Progression, series if the value of Common difference is negative then the Arithmetic Progression series will move towards negative infinity.

The consecutive terms can be also represented in the Arithmetic Progression series as (…, a-3d, a-2d, a-d, a, a+d, a+2d, a+3d).

How to find Arithmetic Progressions

There are certain formulas that can be used to find the Arithmetic Progressions. 

The general formula of “how to find” the Nth term of an Arithmetic progression series is Tn= a + (n-1) d. 

When the question arises of “how to find” the first term of the Arithmetic progression series, it can be written as T1= a+ (1-1) d 

The question of “how to find” the second or third term of the Arithmetic progression series, in place of n the value 2 or 3 need to be written in order to find the required term.

The formula to find out the sum of n terms of Arithmetic Progression is (n/2)[2a+ (n-1)*d]. Suppose in a given series the terms consist of 5,11,17,23. Therefore, a= 5 which is the first term. The common difference d= 11-5= 6 and n the last term is 35. On putting the values in the formula the sum can be determined as 3745. The formula helps in solving the sums   

The question is “how to find” the series by using the reverse order in the Arithmetic Progression. Then the term “l” is considered instead of the first term “a”. The formula of finding the term in reverse order is Sn= [l+ (l-d) + (l-2d) +… (a+2d)+ (a+d)+a].

The formula of finding the sum of such a term in the Arithmetic progression is Sn= (n/2) (a+l) where, as the first term, n is the number of terms in the series, l is the last term of the series.

When the question arises of “how to find” the last term of replacing the last term the formula used for the nth term is a= (n-1) d. In addition to that, the sun can be obtained by [(n/2)(2a+(n-1)d].

In a given series if the first term is given as 5 and the last term is stated as 209 and the n term is given as 35 therefore the formula used to solve the Arithmetic progression series is Sn= 35*2142 which is equal to 3745.

Conclusion

From the above discussions, it can be concluded that the Arithmetic Progression is a sequence of numbers and the consecutive numbers are separated with a common difference. By using the Arithmetic Progression, the finding of sums of longer terms becomes easy in mathematics. Arithmetic progressions help in daily lives and problem-solving takes less amount of time by putting them into simple formulas. The terms consist of common differences, last term, first terms have their own mathematical reparation and formulas.