Question

How do you tell whether the sequence \[3,8,9,12,....\] is arithmetic, geometric or neither?

Answer 1

Hint: Solving arithmetic or geometric sequence problems requires only one thing which is calculating the common difference or the common ratio accordingly. So, here too, we first calculate these and if they come out as constant throughout, we declare the sequence to be arithmetic or geometric or both or neither.

Complete step-by-step answer:
The given sequence is
\[3,8,9,12,....\]
First of all, let us understand what an arithmetic or geometric sequence is. An arithmetic sequence is that in which the difference between the consecutive terms is constant. A geometric sequence is that in which the ratio of the consecutive terms is constant.
Now, to know whether the given sequence is arithmetic or not, we take the difference between the second and first terms which is
\[8-3=5\]
Similarly, we take the difference between the third and second terms which is
\[9-8=1\]
Clearly, the two differences come out to be different. This means that there exists no such common difference between the consecutive terms in the given sequence. So, the sequence is not arithmetic.
Now, to know whether the given sequence is geometric or not, we first take the ratio of the second term to the first term, which is \[\dfrac{8}{3}\] .
Similarly, we take the ratio of the third term to the second term, which is \[\dfrac{9}{8}\] . Clearly, these two ratios are different as well. This means that there exists no such common ratio between the consecutive terms of the given sequence. So, the sequence is not geometric.
Thus, we can conclude that the given sequence is neither arithmetic nor geometric.

Note: In arithmetic and geometric sequence problems, care must be taken while calculating the common difference or the common ratio. We should always remember that the common difference is \[\left( n+1 \right)th\] term – \[nth\] term and the common ratio is \[\dfrac{(\left( n+1 \right)th)term}{(nth)term}\] .