Count Repeated Multiplications in Growth
A doubling model multiplies the current amount by two each period.
About 20 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: A fictional model doubles each period from 20. What is the amount after 3 full periods? Candidate answers: (A) 159 (B) 160 (C) 161 (D) 320.
- Independent check inputs: A fictional model doubles each period from 30. What is the amount after 3 full periods? Candidate answers: (A) 239 (B) 240 (C) 241 (D) 480.
Learn the idea
An exponential pattern changes by a constant multiplicative ratio over equal input steps. This differs from a linear pattern’s constant addition and can model supplied growth, decay or ordered geometric sequence terms. A doubling model multiplies the current amount by two each period. The initial amount is multiplied by a power whose exponent counts the number of completed periods.
Here is the method on the supplied teaching case. The model is A(t)=20·2^t. Three multiplicative steps give 20×2×2×2=160; adding 2 three times would not describe doubling.
The model is supplied rather than a guarantee about a real population. Include period zero as the initial amount; adding the multiplier instead of repeatedly multiplying creates a different pattern.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: A fictional model doubles each period from 20. What is the amount after 3 full periods? Candidate answers: (A) 159 (B) 160 (C) 161 (D) 320. Solution: The model is A(t)=20·2^t. Three multiplicative steps give 20×2×2×2=160; adding 2 three times would not describe doubling. The final answer is 160. The model is supplied rather than a guarantee about a real population. Include period zero as the initial amount; adding the multiplier instead of repeatedly multiplying creates a different pattern.
Check your reasoning
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Helpful tips
- A doubling model multiplies the current amount by two each period.
- The model is supplied rather than a guarantee about a real population.
See the answer and explanation
240
The model is A(t)=30·2^t. Three multiplicative steps give 30×2×2×2=240; adding 2 three times would not describe doubling.
Take it further
Changed-input task: A fictional model doubles each period from 90. What is the amount after 3 full periods? Candidate answers: (A) 1440 (B) 719 (C) 720 (D) 721. Show the mathematical reason, not only the selected answer. Expected result: 720. The model is A(t)=90·2^t. Three multiplicative steps give 90×2×2×2=720; adding 2 three times would not describe doubling.
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More practice: try new cases
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