Question

How to tell whether the sequence $2,4,8,16,.................$ is arithmetic, geometric or neither.

Answer 1

Hint:Arithmetic sequences is a type of sequence where the difference between consecutive terms is always the same.Geometric sequence is a type of sequence where the ratio between consecutive terms is always the same.Now with the help of the above two definitions we can find out whether the given sequence is arithmetic, geometric or neither.

Complete step by step answer:
Given, $2,4,8,16,................................\left( i \right)$
Now our aim is to find out whether the given sequence is arithmetic, geometric or neither.
So let’s take two cases and assume first it to be arithmetic and then geometric to see which of the conditions it satisfies.
Case I:
Assume the given sequence to be an arithmetic sequence.
We have the basic definition of arithmetic sequence as:
Arithmetic sequences:
This is a type of sequence where the difference between consecutive terms is always the same and we call that difference a common difference.
$d = {\text{difference of successive terms}}$
Now let’s take the common difference of the successive terms of the given sequence $2,4,8,16,.................$ such that, we can write:
$d = 4 - 2 = 2 \\
\Rightarrow d = 8 - 4 = 4 \\
\Rightarrow d = 16 - 8 = 8 \\ $
So on observing the above results we can conclude that they do not have a common difference such that the given sequence is not an arithmetic sequence.

Case II:
Assume the given sequence to be a geometric sequence.
We have the basic definition of geometric sequence as:
Geometric sequence:
This is a type of sequence where the ratio between consecutive terms is always the same and we call that ratio a common ratio.
$r = {\text{ratio of successive terms}}$
Now let’s take the common ratio of the successive terms of the given sequence $2,4,8,16,.................$ such that, we can write:
$r = \dfrac{4}{2} = 2 \\
\Rightarrow r = \dfrac{8}{4} = 2 \\
\therefore r = \dfrac{{16}}{8} = 2 \\ $
So on observing the above results we can conclude that they have a common ratio such that the given sequence is a geometric sequence since it satisfies the condition.

Therefore the sequence $2,4,8,16,.................$ is a geometric sequence

Note:General terms of arithmetic and geometric series:
Arithmetic series:${a_n} = {a_{n - 1}} + d$
Geometric series:${a_n} = r \times {a_{n - 1}}$
While doing similar questions one should be very thorough with the properties and formulas regarding the sequences.