A plan doubles its output each period; another produces 100 times the square of the period number. Which eventually produces more? Before answering, write down what “eventually” means: there must be a point after which the comparison stays true. A large value today is not the same as a faster long-run growth mechanism. We use synthetic, unconstrained mathematical models, not a forecast that a real organization can double forever.
Predict
Let E(n)=2^n and P(n)=100n², with integer n≥1. Predict which is larger at n=10 and which is larger at n=16. Record your predictions before calculating. The coefficient 100 can hide the exponential advantage for several periods. Starting at n=0 would also change a question about the first crossing, since P(0)=0; our declared domain deliberately begins at 1.
Experiment
Construct two plots from the exact table below on the same axes. Use n horizontally and output in thousands vertically. First restrict the window to n=4 through 12; then extend it through 16. Explain how the conclusion changes when the evidence window changes.
n
E(n)
P(n)
4
16
1600
8
256
6400
12
4096
14400
14
16384
19600
15
32768
22500
16
65536
25600
These are integer-time models, so separate points are the actual graph. Lines joining them would be visual guides or an additional continuous extension, not additional observed periods. Dividing both outputs by 1000 changes the vertical unit but preserves every comparison.
Understand
At n=14 the exponential is still smaller; at n=15 it is larger. Direct calculation for n=1 through 14 verifies that 15 is the first crossing on our domain. But checking 16 alone does not prove that every later period stays above.
Define R(n)=E(n)/P(n), the ratio of the two positive outputs. Then R(n+1)/R(n)=2n²/(n+1)². For n≥3 this exceeds 1, because 2n²−(n+1)²=n²−2n−1 is positive. Thus R increases from that point onward. Since R(15)>1, every later integer ratio also exceeds 1. This is a proof of persistence, unlike a finite graph.
The same mechanism extends to b^n versus Cn^k for fixed b>1, C>0 and positive integer k. The ratio of consecutive ratios is b(n/(n+1))^k. As n grows, n/(n+1) approaches 1, so this expression approaches b. Choose any q strictly between 1 and b. Eventually every ratio is at least q, and repeated multiplication forces the original ratio to exceed 1 and remain above it. This argument explains why an arbitrarily large fixed coefficient can delay but cannot prevent eventual exponential dominance. It does not supply a practical crossing date without further calculation.
Apply
Compare 3^n with 20n² in the practice questions. Locate a crossing by exact substitution, then justify persistence separately. Also test the boundary assumption b=1: a constant numerator cannot eventually dominate a growing polynomial. If a real process runs out of resources, neither formula may remain credible; distinguish a theorem about a model from evidence that the model still describes the situation.
Continue
Keep three statements separate in your report: what the displayed window shows, what the algebra proves, and what assumptions make the model useful. Next, logarithms turn a positive exponential target into a real-valued time, while the reporting schedule determines how to round that time.
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
For integer n≥1 compare E(n)=3^n and P(n)=20n². What is the first integer n with E(n)>P(n)?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For E(n)=2^n and P(n)=100n², calculate E(10)/P(10).
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Prove that 3^n remains above 20n² for every integer n≥6 after the first crossing.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A graph shows 100n² above 2^n for n=4,…,12. A report says polynomial growth is always faster. Repair the claim using n=14,15 and a distinction between evidence and proof.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Explain whether dividing both plotted outputs by 1000, or starting a graph at n=8, changes the mathematical crossover. Explain the visual risks.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For fixed b>1, C>0 and positive integer k, outline why b^n eventually exceeds Cn^k. Why does this fail for b=1?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
