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NSWMaths Extension 1Quick questions

Proof (ME-P1)

Quick questions on Mathematical induction for series identities

15short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is the principle?
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Let P(n)P(n) be a statement about a positive integer nn. If:
What is the standard four-part structure?
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Part 1: Base case. Verify P(1)P(1) directly by substituting n=1n = 1 into both sides.
What is common series formulas (good to memorise)?
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$βˆ‘i=1ni=n(n+1)2,\sum_{i = 1}^{n} i = \frac{n(n + 1)}{2},$
What is the induction-step algebra?
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The hardest part is the algebra in the inductive step. The pattern:
What is geometric series?
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Prove 1+r+r2+β‹―+rnβˆ’1=rnβˆ’1rβˆ’11 + r + r^2 + \dots + r^{n - 1} = \frac{r^n - 1}{r - 1} for rβ‰ 1r \neq 1.
What is sum of cubes?
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Prove βˆ‘i=1ni3=(n(n+1)2)2\sum_{i = 1}^{n} i^3 = \left( \frac{n(n + 1)}{2} \right)^2.
What is a non-standard sum?
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Prove βˆ‘i=1ni(i+1)=n(n+1)(n+2)3\sum_{i = 1}^{n} i (i + 1) = \frac{n(n + 1)(n + 2)}{3}.
What is part 1: Base case?
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Verify P(1)P(1) directly by substituting n=1n = 1 into both sides.
What is part 2: Inductive hypothesis?
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Assume P(k)P(k) for some positive integer kk. Write the assumed identity explicitly.
What is part 3: Inductive step?
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Use the hypothesis to derive P(k+1)P(k + 1). The standard technique is to write the (k+1)(k + 1)-term sum as the kk-term sum (which the hypothesis gives a formula for) plus the new (k+1)(k + 1)-th term, then simplify to the right form.
What is part 4: Conclusion?
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By the principle of mathematical induction, P(n)P(n) holds for all positive integers nn.
What is assuming what you are trying to prove?
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The hypothesis assumes P(k)P(k), not P(k+1)P(k + 1). Mixing these is circular reasoning.
What is algebra errors in the step?
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Factoring out (k+1)(k + 1) then matching the target form is the standard route. If the algebra is messy, write the target form for n=k+1n = k + 1 and aim at it.
What is missing the conclusion sentence?
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The four-part structure includes a final "by the principle of mathematical induction" sentence. Markers expect it.
What is index confusion?
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βˆ‘i=1n\sum_{i = 1}^{n} goes from i=1i = 1 to i=ni = n. A (k+1)(k + 1)-term sum includes the term at i=k+1i = k + 1, which is the new term you add in the step. :::

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