Question

How do you determine whether the sequence \[1,\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},...\] is geometric and if it is, what is the common ratio?

Answer 1

Hint: First understand the general form of geometric progression. Assume the given four terms as \[{{T}_{1}},{{T}_{2}},{{T}_{3}},{{T}_{4}}\] respectively. Now, to check if the sequence is a G.P. or not, find the values of the expression \[\dfrac{{{T}_{2}}}{{{T}_{1}}},\dfrac{{{T}_{3}}}{{{T}_{2}}},\dfrac{{{T}_{4}}}{{{T}_{3}}}\]. If all these three ratios give the same value then the sequence is a G.P. otherwise not. The common ratio will be \[r=\dfrac{{{T}_{2}}}{{{T}_{1}}}=\dfrac{{{T}_{3}}}{{{T}_{2}}}=\dfrac{{{T}_{4}}}{{{T}_{3}}}\] if and only if the sequence turns out to be G.P.

Complete step by step answer:
Here, we have been provided with four terms of a sequence \[1,\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},...\] and we have to determine if the sequence is a G.P. or not. If it is a G.P. then we have to find the common ratio. But first, let us know about the geometric progression (G.P.).
1. Geometric Progression: - A G.P. is a sequence in which the successive term differs from its previous term by a fixed ratio called the common ratio. General form of the G.P. is given as: - \[a,ar,a{{r}^{2}},a{{r}^{3}},.....,a{{r}^{n-1}}\]. Here, ‘a’ is the first term and ‘r’ is the common ratio.
Now, let us come to the question. Let us assume the four terms provided as \[{{T}_{1}},{{T}_{2}},{{T}_{3}}\] and \[{{T}_{4}}\] respectively, so we have,
\[\begin{align}
  & \Rightarrow {{T}_{1}}=1 \\
 & \Rightarrow {{T}_{2}}=\dfrac{1}{2} \\
 & \Rightarrow {{T}_{3}}=\dfrac{1}{3} \\
 & \Rightarrow {{T}_{4}}=\dfrac{1}{4} \\
\end{align}\]
Let us check if these terms are in G.P. or not. So, we have,
\[\Rightarrow \dfrac{{{T}_{2}}}{{{T}_{1}}}=\dfrac{\dfrac{1}{2}}{1}=\dfrac{1}{2}\]
\[\Rightarrow \dfrac{{{T}_{3}}}{{{T}_{2}}}=\dfrac{\dfrac{1}{3}}{\dfrac{1}{2}}=\dfrac{2}{3}\]
\[\Rightarrow \dfrac{{{T}_{4}}}{{{T}_{3}}}=\dfrac{\dfrac{1}{4}}{\dfrac{1}{3}}=\dfrac{3}{4}\]
It is clear from the above results that there is no common ratio between the successive terms. So, we can conclude that the sequence is not a G.P.

Note:
One may note that if we are asked what type of sequence is it given in the question then we would have concluded that the sequence is a harmonic progression (H.P.). This is because we can see that the reciprocal of the given terms will form an arithmetic progression. You must remember all the types of sequences and their properties as anyone can be asked to check.