How can I tell exponential growth from linear growth in a table?

Start by checking the inputs: they need to be equally spaced for a straightforward comparison between consecutive rows. For each pair of neighboring outputs, calculate both the difference (later output minus earlier output) and the ratio (later output divided by earlier output). A constant difference points to a linear pattern; a constant ratio points to an exponential pattern, because the outputs respectively add a fixed amount or multiply by a fixed factor. [S1]

For example, if the inputs go 0, 1, 2, 3, compare each neighboring pair in that order. Don’t decide based only on the first two outputs: two patterns can share the same starting values and then diverge. [S1]

Assume the inputs increase by 1 each time. Here are two patterns, with the calculations for each pair of consecutive outputs.

Linear pattern

Input       0    1    2    3
Output    100  150  200  250
Difference  —  +50  +50  +50
Ratio       —  1.5  1.33… 1.25

The difference stays +50, so this table shows linear growth. The ratios change, so they are not the consistent pattern here.

Exponential pattern

Input       0    1    2      3
Output    100  150  225  337.5
Difference  —  +50  +75  +112.5
Ratio       —  1.5  1.5    1.5

The ratio stays 1.5, so this table shows exponential growth. Its differences change. These are worked examples: the first adds 50 each step, while the second multiplies by 1.5 each step. [S1]

If the inputs are evenly spaced by a fixed interval other than 1, compare outputs over that same interval; don’t treat uneven steps as equivalent consecutive steps. The same idea—checking a repeated change or repeated factor over equal input intervals—applies. [S1]

A zero output makes a ratio from that row undefined, because division by zero is not allowed. So you can’t use that particular ratio to test the table. Look at other valid consecutive ratios, and also calculate the differences. If the available rows don’t provide a clear, consistent pattern, mark the result as uncertain rather than forcing a classification.

Show the input spacing, the calculations, and the conclusion. For example, for the second table you could write:

“Inputs increase by 1 each time. The consecutive output ratios are 150 ÷ 100 = 1.5, 225 ÷ 150 = 1.5, and 337.5 ÷ 225 = 1.5. Since the ratio is constant, the outputs follow an exponential pattern over these rows.”

For the first table, replace the ratio calculations with the differences: 150 − 100 = 50, 200 − 150 = 50, and 250 − 200 = 50. Then say the constant difference indicates a linear pattern. A finite table can show a pattern without uniquely proving the rule for every possible input; the conclusion should describe the observed pattern and any assumptions your assignment gives. [S1]