Distinguish a tested pattern from an induction argument with adequate base cases.
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Try
A recurrence or sum often looks correct for several small indices. Induction turns a repeatable step into a proof, but only if the step starts from established cases and reaches every intended index.
Predict
Predict how many starting cases are needed when your step goes back three places. A one-step induction usually needs one base. A three-step reduction can skip entire residue classes unless the bases cover all of them.
Experiment
Use the recurrence investigation to compare accumulation with a proposed formula. Then sketch arrows from each integer n≥11 back to n−3. Follow the chains to 8,9 or 10. This visual explains the base block; it does not by itself prove that the algebraic property is preserved.
Understand
Worked argument: for every nonnegative integer n, 1+3+…+(2n−1)=n², with the empty sum at n=0 defined as 0. The base is 0=0. Assume the formula at n. Adding the next odd number 2n+1 gives n²+2n+1=(n+1)². Thus the formula holds at the successor, proving all nonnegative indices.
An invalid proof of a step-three theorem checks only n=8, then adds 3 repeatedly. It reaches 8,11,14 and so on, but never reaches 9 or 10. The induction step may be correct while the coverage is incomplete.
Apply
In the summation task, identify exactly which term is newly added. In the representation task, state the full induction hypothesis and verify that n−3 belongs to its range. Check nonnegative coefficients after constructing the next representation.
Continue
Depth: strong induction and ordinary induction are logically equivalent: an ordinary induction statement can assert that all cases up to n hold. The stronger hypothesis is a convenient organization, not permission to assume the target itself. Distinguish proving a recurrence formula from merely using it to predict a later value.
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Practice and reasoning
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.
Compute the sum of k(k+1) for k=1,2,3,4.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
To prove every n≥8 is 3a+5b using a step from n−3, which base block suffices?
Prove Σ(k=1..n) k(k+1)=n(n+1)(n+2)/3 for every positive integer n.
Structure your argument and preserve revisions
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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.
Prove every integer n≥8 can be written 3a+5b with nonnegative integers a,b.
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.
