Question

How do you know if the sequence 1, -2, 4, -8,……. is arithmetic or geometric?

Answer 1

Hint: First of all understand the general forms of geometric and arithmetic progressions. Assume the given four terms as \[{{T}_{1}},{{T}_{2}},{{T}_{3}}\] and \[{{T}_{4}}\] respectively. Now, to check if the sequence is an A.P., find the values of the expressions \[{{T}_{2}}-{{T}_{1}},{{T}_{3}}-{{T}_{2}},{{T}_{4}}-{{T}_{3}}\]. If all these three expressions give the same value then the sequence is an A.P. Now, to check for the G.P., find the values of the expressions \[\dfrac{{{T}_{2}}}{{{T}_{1}}},\dfrac{{{T}_{3}}}{{{T}_{2}}}\], \[\dfrac{{{T}_{4}}}{{{T}_{3}}}\]. If all these three expressions give the same value then the sequence is a G.P.

Complete step by step answer:
Here, we have been provided with four terms of a sequence 1, -2, 4, -8, ….. and we have to determine if the sequence is arithmetic or geometric. But first, let us know about the arithmetic progression (A.P.) and geometric progression (G.P.).
(1) Arithmetic progression: - An A.P. is a sequence in which the successive term differs from its previous term by a fixed number called the common difference. General form of an A.P. is given as: - a, a + d, a + 2d, a + 3d,….., a+ (n – 1)d. Here, ‘a’ is the first term and ‘d’ is a common difference. For example: - 10, 20, 30, 40,…. are in A.P. with the common difference as 10.
(2) 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 \[T,{{T}_{2}},{{T}_{3}}\] and \[{{T}_{4}}\] respectively, so we have,
\[\begin{align}
  & \Rightarrow {{T}_{1}}=1 \\
 & \Rightarrow {{T}_{2}}=-2 \\
 & \Rightarrow {{T}_{3}}=4 \\
 & \Rightarrow {{T}_{4}}=-8 \\
\end{align}\]
First let us check if these terms are in A.P. or not, so we get,
\[\begin{align}
  & \Rightarrow {{T}_{2}}-{{T}_{1}}=-2-1=-3 \\
 & \Rightarrow {{T}_{3}}-{{T}_{2}}=4-\left( -2 \right)=6 \\
 & \Rightarrow {{T}_{4}}-{{T}_{3}}=-8-\left( 4 \right)=-12 \\
\end{align}\]
Clearly, we can see that there is not a common difference between the successive terms, so we can say that the sequence is not an A.P.
Now, let us check if these are in G.P. or not, so we get,
\[\begin{align}
  & \Rightarrow \dfrac{{{T}_{2}}}{{{T}_{1}}}=\dfrac{-2}{1}=-2 \\
 & \Rightarrow \dfrac{{{T}_{3}}}{{{T}_{2}}}=\dfrac{4}{-2}=-2 \\
 & \Rightarrow \dfrac{{{T}_{4}}}{{{T}_{3}}}=\dfrac{-8}{4}=-2 \\
\end{align}\]
It is clear from the above results that there is a common ratio between the successive terms, so we can conclude that the sequence is a G.P.

Note:
One must remember the definitions and general forms of these sequences, A.P. and G.P., to solve the above question. You may see that there will be other terms also after -8 but we do not have to consider them in our solution. However, you can find the terms after it is known what type of sequence is given. We need at least four terms to determine the sequence. Remember that there is one more sequence called harmonic progression (H.P.) in which the reciprocal of the terms are in A.P., so you must remember this sequence also.