Linear and Geometric Growth: Fresh Take

  • Identify whether a scenario or data describes linear or geometric growth
  • Identify key growth parameters, such as growth rates and initial values, and express them in a format that can be used for calculation
  • Use equations to predict future values for linear and exponential growth

Predicting Growth

The steps of determining the formula and solving the problem of Marco’s bottle collection shown on the Learn It page 1 are explained in detail in the following videos.

You can view the transcript for “Linear Growth Part 1” here (opens in new window).

You can view the transcript for “Linear Growth Part 2” here (opens in new window).

You can view the transcript for “Linear Growth Part 3” here (opens in new window).

Linear Growth

The Main Idea

linear growth

If a quantity starts at size [latex]P_0[/latex] and grows by [latex]d[/latex] every time period, then the quantity after [latex]n[/latex] time periods can be determined using either of these relations:

Recursive form

[latex]P_n = P_{n-1} + d[/latex]

Explicit form

[latex]P_n = P_0 + d n[/latex]

In this equation, [latex]d[/latex] represents the common difference – the amount that the population changes each time [latex]n[/latex] increases by [latex]1[/latex].

The cost, in dollars, of a gym membership for [latex]n[/latex] months can be described by the explicit equation [latex]P_n = 70 + 30_n[/latex]. What does this equation tell us?

Exponential (Population) Growth

The Main Idea

exponential growth

If a quantity starts at size [latex]P_0[/latex] and grows by [latex]R\%[/latex] (written as a decimal, [latex]r[/latex]) every time period, then the quantity after [latex]n[/latex] time periods can be determined using either of these relations:

Recursive form

[latex]P_n = (1+r)P_{n-1}[/latex]

Explicit form

[latex]P_n = (1+r)^{n}P_0[/latex]         or equivalently, [latex]P_n= P_0(1+r)^{n}[/latex]

We call [latex]r[/latex] the growth rate.

The term [latex](1+r)[/latex] is called the growth multiplier, or common ratio.

Between 2007 and 2008, Olympia, WA grew almost [latex]3\%[/latex] to a population of [latex]245[/latex] thousand people. If this growth rate was to continue, what would the population of Olympia be in 2014?

Watch the following videos for more on linear vs. exponential growth.

You can view the transcript for “Examples of linear and exponential relationships” here (opens in new window).

You can view the transcript for “Understanding linear and exponential models | Functions and their graphs | Algebra II | Khan Academy” here (opens in new window).

You can view the transcript for “Linear vs. exponential growth: from data | High School Math | Khan Academy” here (opens in new window).