A Short Note On Progressions

Arithmetic Progression is a series of two or more numbers that are associated with the adjacent number itself by some relation. The relation depicts that the two numbers have a common difference from each other. The difference is not any random difference, but the difference is always constant between any member of an arithmetic progression. The common difference between numbers is denoted by “d”.  For example, a series of even natural numbers less than 20 is in arithmetic progression because the common difference is 2. It doesn’t matter whatever way you want to check the difference. Arithmetic progression formulas are pre-defined sets of axioms that can be used to reduce the complication of the questions and retrieve the correct answer easily. 

Difference Between Sequence & Progression

Arithmetic Progression is one of the types of series and sequences available in the bucket of mathematics algebra. This branch of maths is related to number systems and algebraic operations are a major part of it. Some of the other series and progressions are harmonic progressions and geometric progressions. 

There is a difference between a sequence and a progression. This difference is little which means that a progression will always be in a sequence, but a sequence does not have to be a progression all the time. 

A sequence is just an order of a few numbers that are arranged in any random manner together but a progression is not any random arrangement of numbers. Every element of a progression is placed somewhere in the arrangement because it holds a significant role, If you’ll observe the progression deeply, you can see a common pattern that is repeating itself. This common pattern is called the nth term of that progression.

What Is Arithmetic Progression?

Arithmetic Progression has progression which also means that arithmetic progression is a type of sequence which is also a progression because it has a common pattern that repeats itself after every number. We already discussed what is a common difference in the above paragraph. 

Arithmetic Progression can be explained as the mathematical sequence of numbers whose terms are different from each other; every term and its adjacent term from any side left or right shares a constant gap or bridge that we call a common difference. 

Terminologies In Arithmetic Progression

1. Common difference (the gap/bridge): – The arithmetic progression relation depicts that the two numbers have a common difference from each other. The difference is not any random difference, but the difference is always constant between any member of an arithmetic progression. The common difference between numbers is denoted by “d”.
2. The general term of an AP(nth term): – The general pattern that we’ve talked about in the above section is what a general tern or the nth term of an arithmetic progression means. 
   an = a + (n − 1) * d 

where a is the first number or term of the arithmetic progression.

n is the number of terms or numbers in the arithmetic progression.

  3.Sum of the n numbers of an AP

The sum of n numbers of an AP is something that we haven’t discussed before. You can get an idea that it is the aggregate of all the consecutive terms till a certain term i.e. nth. 

S = n/2{2a + (n − 1) * d}

where a is the first number or term of the arithmetic progression.

n is the number of terms or numbers in the arithmetic progression.

#Note: – If in an AP, the first number and last number are already provided in the series, then you can calculate the sum of n terms by this simple and easy formula:

S  = n/2 (a + l), where a is the first number and l is the last number.

Geometric Progression

Geometric Progression is also one of the progressions that are similar to arithmetic progression but there are a few differences. In AP, we got that there is a term called “common difference” but here in geometric progression, there is no term like the common term but there is “common ratio”. Geometric Progression is the play of calculating and finding equal common ratios between two adjacent terms. 

Geometric Progression has a common ratio ad we know that ratios are nothing but dividends. The ratio is the fraction when one number has to get divided by some other number. The degree of the second term in the geometric progression is everything. The degree or the power depicts the common ratios between the nth terms of the GP.

Harmonic Progressions

Harmonic Progression is the equivalent of arithmetic progression. Harmonic progression is the progression where there is a “common difference” between their adjacent nth terms. This common difference is the same common difference as in arithmetic progression. 

The reason that makes harmonic progression and arithmetic progression different from each other is the nth term formula. 

If the AP is 3,6,9,12,__.

Then ,we know 

6-3=9-6=12-9=3 i.e. the common difference

So, the next term will be 12 + 3 = 15.

In the case of HP, 

1/2,1/4, 1/6, 1/8, 1/16

You can notice that if you write the denominators separately, they would fit in the format of AP.

Hence, harmonic progressions terms are the terms whose denominators are in the arithmetic progression in a proper sequence with the same common difference.

Conclusion

Arithmetic Progression, or AP for short, is a mathematical concept that describes to a succession of integers in a certain or particular order. We have seen many occasions to witness the use of arithmetic progression. The arithmetic operation can allow us to find the second number in an arithmetic progression by adding the constant number to the previous number and finding many more numbers consecutively.