Question

Is the sequence \[4,{\text{ }}16,{\text{ }}36,{\text{ }}64,.....\] arithmetic?

Answer 1

Hint: In this question we have to check whether the given sequence is in arithmetic progression or not. For this, we will find the difference between each of the two consecutive terms to check whether the difference is constant or not by using the common difference formula which is given by: \[d = {a_{n + 1}} - {a_n}\] . So, if the difference turns out to be constant, then we can say that the given sequence is in A.P. otherwise it is not in A.P.

Complete step by step answer:
The given sequence is:
\[4,{\text{ }}16,{\text{ }}36,{\text{ }}64,.....\]
And we have to check whether the given sequence is in arithmetic progression or not.
In order to check whether this sequence is arithmetic, we need to calculate the common difference between consecutive terms.
So, let’s find out the common difference.
Let the first term be \[{a_1}\]
Hence, \[{a_1} = 4\]
Let the second term be \[{a_2}\]
Hence, \[{a_2} = 16\]
Let the third term be \[{a_3}\]
Hence, \[{a_3} = 36\]
and so on.
Now, we know that
Common difference, \[d = {a_{n + 1}} - {a_n}\]
So, let us find the difference between first and the second term:
i.e., \[d = {a_2} - {a_1}\]
on putting the respective values, we get
\[d = 16 - 4\]
\[ \Rightarrow d = 12{\text{ }} - - - \left( i \right)\]
Now, let us find the difference between second and the third term:
i.e., \[d = {a_3} - {a_2}\]
on putting the respective values, we get
\[d = 36 - 16\]
\[ \Rightarrow d = 20{\text{ }} - - - \left( {ii} \right)\]
Since the value of \[d\] in equation \[\left( i \right)\] and equation \[\left( {ii} \right)\] is not the same.
Therefore, we can conclude that the difference is not constant.
Hence, the given sequence is not an arithmetic progression.

Note:
Whenever you face such types of problems, you need to know the concept of arithmetic progression and geometric progression. In geometric progression we will always check common ratios and that should be constant to be in GP.
Formula of $n^{th}$ term of GP = $ar^{n-1}$