Linear Versus Exponential Change

A linear function like f(x) = 100 + 50x grows by repeated addition: each time x goes up by 1, you add 50 more. That constant addition is what makes the graph a straight line with slope 50. An exponential function like g(x) = 100 times 1.5^x grows by repeated multiplication: each time x goes up by 1, you multiply the current value by 1.5. Multiplying the same growing number over and over means the increase itself keeps increasing, which bends the graph upward into a curve.

Compare the two at x = 0, 5, 10, 20. At x = 0 both equal 100. At x = 5, f(5) = 350 but g(5) is about 759. At x = 10, f(10) = 600 while g(10) is about 5767. At x = 20, f(20) = 1100, while g(20) has ballooned past 332,000. The linear function never changes its pace. The exponential function's pace keeps accelerating, so no matter how steep you make a line, a genuine exponential eventually blows past it. That is why the doubling penny beats a flat million dollars within a month: thirty doublings turns one cent into over five million dollars, because each day's gain depends on the day before, not on a fixed amount.

This distinction is also why arithmetic sequences and linear functions are the same idea in different clothing: an arithmetic sequence adds a common difference each term, exactly like a linear function adds a constant slope each step. Geometric sequences and exponential functions pair up the same way, through a common ratio instead of a common difference. Recognizing which mechanism, addition or multiplication, is driving a data set is the real skill: a table where equal x-steps give equal differences in y is linear, while equal x-steps giving equal ratios in y is exponential.