Question

Find out if the given below sequence is an arithmetic progression. If so then calculate the common difference.
3, 6, 12, 24, …..

Answer 1

Hint:To solve this question we will, first of all, define AP (Arithmetic Progression). If the terms of the sequence are given as a, a + d, a + 2d, a + 3d, …. etc. then the sequence is an AP with a common difference as ‘d’. We will try to relate this with the given sequence and hence check if it is an AP or not.

Complete step by step answer:
In the case of an Arithmetic Progression, some special points are to be taken care of. Let us first define an Arithmetic Progression.
An Arithmetic Progression is a sequence of numbers such that the difference of any two numbers, the successive number is a constant. The short form of Arithmetic Progression is AP. For example,
(1) 1, 2, 3, 4, 5, 6 .... is an AP of common difference 1.
(2) 3, 5, 7, 9, 11….. is an AP of common difference 2.
We are given our sequence as 3, 6, 12, 24….
Observer that in an AP, the common difference between each successive number is the same. So, here,
\[6-3=3\]
\[12-6=6\]
As, \[3\ne 6\] so,
\[\Rightarrow 6-3\ne 12-6\]
So, the common difference between the successive terms is not the same.
Hence, the given square 3, 6, 12, 24 is not an AP.

Note:
If we add certain terms between the terms of the sequence 3, 6, 12, 24, then it can surely become an AP. Just one point to keep note that the common difference is the same. 3, 6, 9, 12, 15, 18, 21, 24…. is an AP as the common difference is 3. In this series, the common ratio is the same though. We have,
\[\dfrac{6}{3}=2\]
\[\dfrac{12}{6}=2\]
\[\dfrac{24}{12}=2\]
So, here, we can see that the common ratio is 2. We know that we have a Geometric Progression in which the terms have a common ratio between them. So this is in fact a GP series.