Compute the ratios of successive entries in the table to...

Compute the ratios of successive entries in the table to determine whether or not an exponential model is appropriate for the data.
$$\begin{array}{|l|l|l|l|l|l|l|}
\hline x & 0 & 2 & 4 & 6 & 8 & 10 \\
\hline y & 3 & 15.2 & 76.9 & 389.2 & 1975.5 & 9975.8 \\
\hline
\end{array}$$

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Key Concepts

Exponential Model

An exponential model is used to describe data that increases or decreases by a constant factor over equal intervals of the independent variable. It is usually expressed in the form y = ab^x, where a represents the initial amount and b is the constant base or growth factor.

Ratios of Successive Terms

Computing the ratios of successive terms in a dataset involves dividing each term by the previous term. This process helps reveal whether the data exhibit a consistent multiplicative change, which is characteristic of exponential behavior.

Constant Ratio Verification

If the ratios obtained from successive entries remain consistently equal (or very close to one another), it indicates that the data are increasing by a constant factor. This is a key sign that an exponential model may be an appropriate representation of the data.

Model Appropriateness

Determining the appropriateness of an exponential model involves checking if the observed data consistently adhere to the structure of exponential growth (or decay). A constant ratio among successive data points is a primary indicator that using an exponential model can be justified.

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