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Outbyte PC Repair FREERepair Windows errors before they cause bigger problemsFix Now →Outbyte Driver Updater FREEScan for outdated or missing drivers - takes under a minuteDriver Scan →The power of doubling is the pattern you get when you repeatedly multiply a starting amount by 2. After n doublings, a starting value A becomes A × 2n. It describes exponential growth; it is not a single separate operation with one uniquely formal definition.
How repeated doubling works
One doubling multiplies a value by 2: doubling x gives 2x. Each further doubling multiplies the latest result by 2 again. Starting from 3, for example, the values are:
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- Starting value, before any doublings: 3
- After 1 doubling: 6
- After 2 doublings: 12
- After 3 doublings: 24
The sequence is geometric, with a common ratio of 2: each term is twice the term before it. The New York State Common Core Algebra I lesson describes this repeated-multiplication pattern as exponential growth (Lesson 5: The Power of Exponential Growth).
Formula for the power of doubling
If A is the amount before any doubling has taken place and n is the number of completed doublings, use:
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Amount after n doublings = A × 2n
For example, starting with 5 and doubling it 4 times gives 5 × 24 = 80. The exponent counts the number of times the value has been multiplied by 2.
When the first term is numbered 1
Some sequences call the starting amount the first term rather than step zero. If term 1 is A, then term k is:
Term k = A × 2k−1
For instance, a chessboard sequence with 1 grain on square 1 has 2 grains on square 2 and 4 on square 3. Its term for square k is 2k−1, because reaching square 1 required no doublings. The New York State lesson presents this one-based convention alongside the general formula for a different starting value.
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Doubling is exponential, not linear growth
The expressions 2n and 2n describe different patterns:
| Rule | What changes at each step? | Values for n = 1, 2, 3, 4 |
|---|---|---|
| 2n (linear) | Add 2 | 2, 4, 6, 8 |
| 2n (exponential) | Multiply by 2 | 2, 4, 8, 16 |
With linear growth, equal increases in the input add the same amount. With exponential growth, each equal step multiplies the result by the same factor. New Zealand Ministry of Education teaching material also illustrates powers of two as successive doublings, including 20 = 1 and 21 = 2 (Powerful Numbers).
Example: a penny that doubles daily
Suppose day 1 starts at $0.01 and the amount doubles once each day after that. On day k, the amount is $0.01 × 2k−1. On day 30, that gives $5,368,709.12.
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That is the amount on day 30, not the total received across all 30 days. Adding the daily amounts from day 1 through day 30 gives $10,737,418.23. These figures use the day-1 starting convention described in the educational handout APES Lab: The Power of Doubling, revised edition 2007.
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How to calculate a doubling problem
- Identify the starting amount. Use the value before any doublings, or note whether the problem calls it term 1.
- Count completed doublings. If the starting value is at day 1, day 30 includes 29 doublings; if it is at day 0, day 30 includes 30.
- Apply the formula. Multiply the starting value by 2 raised to the number of completed doublings.
- Check what the question asks for. A final-day amount is one term. A total across days or steps requires adding the terms, not reporting only the last one.
What the phrase means—and does not mean
In a question about repeated growth, “power of doubling” means the effect of repeated multiplication by 2, often represented with powers of 2 or an exponential formula. The phrase itself is descriptive, rather than the name of a uniquely defined mathematical operation. If the context is geometry—for example, doubling a cube—the question concerns a different problem and needs its own interpretation.
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