Question

Is the given progression arithmetic progression? Why?
2, 3, 5, 7, 8, 10, 15, ….

Answer 1

Hint: Arithmetic Progression (A.P.) is a sequence of numbers in which the differences between any two consecutive terms is a constant.
For example, consider a sequence 1, 3, 5, 7, …
The difference between any two consecutive terms here is 2. So, it can be considered as an A.P.

Complete step-by-step answer:
The given progression is 2, 3, 5, 7, 8, 10, 15, …. and we are asked to find whether it is an Arithmetic progression or not.
For a progression to be A.P. the common difference d between the consecutive terms must be same.
 $d = {a_n} - {a_{n - 1}}$
Here, ${d_1} = {a_2} - {a_1} = 3 - 2 = 1$ and ${d_2} = {a_3} - {a_2} = 5 - 3 = 2$ .
It is clear that the common differences ${d_1}$ and ${d_2}$ between the consecutive terms are not the same.
Thus, the given progression is not an arithmetic progression.

Additional Information:
Types of arithmetic progressions:
There are mainly two types of A.P. i.e., finite and infinite. Where,
If the number of terms in the A.P. are finite i.e. countable then the A.P. is called finite A.P. and if the number of terms in the A.P. are not finite i.e. not countable then such A.P. is called an infinite A.P.
Moreover, The A.P. can be increasing and decreasing; if the value of common difference ‘d’ is positive then it is called increasing A.P> and if the value of common difference ‘d’ is negative then it is called decreasing A.P.
Hence, the given question shows the increasing infinite A.P.

Note: For a progression to be arithmetic, the common difference ‘d’ between the consecutive terms must be the same. Hence, the given sequence is an arithmetic progression or not that can be decided after getting the value of common difference for the same.