Quick questions on Proof by mathematical induction (SACE Stage 2 Specialist Mathematics)

Q: What is inductive step?

Assume P(k) holds for some integer k 1, that is $1+2++k=(k(k+1))/(2).We must showP(k+1): that1+2+\cdots+(k+1)=\dfrac{(k+1)(k+2)}{2}$.

3short 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 base case?

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Left side

=1

. Right side

=\dfrac{1(1+1)}{2}=1

. So

P(1)

is true.

What is inductive step?

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Assume

P(k)

holds for some integer

k\ge 1

, that is $

1+2+\cdots+k=\frac{k(k+1)}{2}.

We must show

P(k+1)

: that

1+2+\cdots+(k+1)=\dfrac{(k+1)(k+2)}{2}$.

What is conclusion?

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By the principle of mathematical induction, the formula holds for all integers

n\ge 1

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