Question

What is progression?

Answer 1

Hint: We try to state a definition or idea about what progression is. Then we gave ideas about different types of progression that have different characteristics. We try to understand the concept with examples of every type.

Complete step-by-step answer:
Progression or series is generally the sequence of numbers which are dependent on some predetermined rules or conditions. There has to be a relation of singular or multiple binary operations among the numbers. The first number of the sequence which is called initial number is random, but after that every number following is condition based which can’t be random.
There are different types of progressions. We can divide them according to their continuations. There are 2 types of series – finite and infinite. If the series ends after a certain value then it is a finite progression and if it continues till infinity then it’s called infinite series. For example: the series $1,2,3,4,....,n-1,n$ is a finite series as there are countable (n) number of terms in the series. But $1,2,3,4,....\infty $ represents an infinite number of terms. There are uncountable numbers of terms in the series.
We can also categorise them according to the operations involved in the series. Those types are Arithmetic progression, Geometric progression, Harmonic progression. Arithmetic progression involves common differences between any two consecutive numbers of that series. So, the binary operation involved in this case is mainly addition of a fixed number to any number of the series for getting the following number of that series. For example: the series $3,8,13,18,.....,5n-2$ is an example of A.P. where the common difference is 5, the difference between any two consecutive numbers. The general term of the series is $5n-2$. The second type of progression is Geometric progression. It involves binary operation of multiplicity. A fixed number is multiplied to any number of the series for getting the following number of that series. The ratio of any two consecutive numbers of that series is constant and is called the common ratio. For example: the series $3,9,27,81,.....,{{3}^{n}}$ is an example of G.P. where the common ratio is 3, the multiplier between any two consecutive numbers.
The third one is Harmonic progression which is actually inverse of A.P. This means if the inverse terms of all the numbers of a series is in A.P. then the numbers of the original series are in H.P. For the previous taken examples of the A.P. $3,8,13,18,.....,5n-2$ the H.P. form will be $\dfrac{1}{3},\dfrac{1}{8},\dfrac{1}{13},\dfrac{1}{18},.....,\dfrac{1}{5n-2}$. All the three types of progression can have both finite and infinite series.

Note: We need to remember that we can make any kind of progression out of the box. It just needs to follow a fixed rule to get the terms of that series. The series ${{3}^{n}}+5n-2$ is an A.G.P series which follows both A.P and G.P.