Question

What is the equation of the data? And how did you get it?
$\begin{align}
  & x=0,1,2,3,4 \\
 & y=2,6,18,54,162 \\
\end{align}$

Answer 1

Hint: In this question, we have to find the equation from the given data. Thus, we will use an arithmetic-geometric sequence formula to get the solution. First, we see that the values of x are in the form of an arithmetic sequence, where the first term is equal to 0, the difference between the consecutive terms is 1 and the total number of terms is 5, thus we get a new equation of n in terms of y. After that, we know that the value of y is in the form of a geometric sequence, thus we will apply the formula ${{a}_{n}}$ , to get the value of y in terms of n. In the last, we will substitute the value of n is the equation of y, to get the required solution for the problem.

Complete step by step solution:
According to the problem, we have to find the equation from the given data.
Thus, we will use the arithmetic-geometric formula to get the solution.
The data given to us is
$\begin{align}
  & x=0,1,2,3,4 \\
 & y=2,6,18,54,162 \\
\end{align}$
Now, we know that the values of x are in the form of arithmetic sequence, where $a=0$ , the difference between any consecutive terms is equal to $d={{a}_{2}}-{{a}_{1}}=1-0=1$ , and let us suppose the last term ${{a}_{n}}=x$ . So, we will put these value in the formula ${{a}_{n}}=a+\left( n-1 \right)d$ , we get
$\Rightarrow x=0+\left( n-1 \right)1$
Therefore, we get
$\Rightarrow x=n-1$
Now, we will add 1 on both sides in the above equation, we get
$\Rightarrow x+1=n-1+1$
As we know, the same terms with opposite signs cancel out each other, we get
$\Rightarrow x+1=n$ -------- (1)
Also, we know that the values of y is in the form of geometric sequence, where $a=2$ and the ratio between any two consecutive terms is equal to $r=\dfrac{{{a}_{2}}}{{{a}_{1}}}=\dfrac{6}{2}=3$ . Let us suppose the last term is equal to ${{a}_{n}}=y$ . Thus, we will put this value in the formula ${{a}_{n}}=a{{r}^{n-1}}$ , we get
$\Rightarrow y={{2.3}^{n-1}}$
Now, we will put the value of equation (1) in the above equation, we get
$\Rightarrow y={{2.3}^{x+1-1}}$
As we know, the same terms with the opposite signs in the power cancel out each other, we get
$\Rightarrow y={{2.3}^{x}}$ which is the required solution.

Therefore, the equation from the given data is equal to $y={{2.3}^{x}}$

Note: While solving this problem, do the step by step calculations to avoid mathematical error. Do mention the arithmetic and the geometric sequence formula to get an accurate answer. You can check your answer by substituting the value of x in the solution to get the value of y.