Question

How do you determine whether the sequence $1,2,4,8,16,....$ is arithmetic and if it is, what is the common difference?

Answer 1

Hint: We have been asked to verify whether the given sequence $1,2,4,8,16,....$ is arithmetic or not. From the basic concepts we know that the terms of arithmetic sequence have common differences. So here we need to verify the difference between different consecutive terms.

Complete step-by-step solution:
Now considering from the question we have been asked to verify whether the given sequence $1,2,4,8,16,....$ is arithmetic or not.
From the basic concepts of sequences we know that the consecutive terms of arithmetic sequence has common differences.
So here we need to verify the difference between different consecutive terms.
We will verify the difference of $1,2$ , $2,4$ , $4,8$ and $8,16$ . The difference between $1,2$ is $\Rightarrow 2-1=1$ . Similarly the difference between $2,4$ is $\Rightarrow 4-2=2$.
These consecutive terms do not have a common difference so they are not in arithmetic sequence.
If we observe the ratios of the consecutive terms are equal. Therefore these consecutive terms have a common ratio.
Hence we can conclude that the given terms are in geometric sequence.

Note: This type of questions are very simple, involve less calculations, very few mistakes are possible and can be solved in a less span of time. We can also find the ${{n}^{th}}$ term of the geometric sequence by using the formula given as $a{{r}^{n-1}}$ where $a$ is the first term and $r$ is the common ratio. For the given sequence the ${{n}^{th}}$ term will be given as $\Rightarrow \left( 1 \right){{\left( 2 \right)}^{n-1}}$ . Similarly we can find the ${{n}^{th}}$ term of the arithmetic sequence by using the formula given as $a+\left( n-1 \right)d$ where $a$ is the first term and $d$ is the common difference.