Proof by mathematical induction — Question 16

QCAA 2024, Paper 1 · 6 marks

Q16 · 2024 · Technology-freeComplex familiar6 marks

QUESTION 16 (6 marks)

Use mathematical induction to prove that

12^n+2(5^{n-1})

is a multiple of 7 for

n\in\mathbb Z^+

WORKED SOLUTION

6 marks

ANSWER

The statement holds for all

n\in\mathbb Z^+

Worked solution

For

n=1

12^1+2(5^0)=14=7\cdot2

. Assume for

n=k

that

12^k+2(5^{k-1})=7m

for some positive integer

m

. For

n=k+1

12^{k+1}+2(5^k)=12(12^k)+10(5^{k-1}).

Rewrite this as

12[12^k+2(5^{k-1})]-14(5^{k-1}),

then use the assumption to obtain

7[12m-2(5^{k-1})],

which is a multiple of 7. Therefore the proposition is true for all

n\in\mathbb Z^+

by mathematical induction.

correctly proves the initial statement

[1 mark]

correctly formulates an appropriate assumption

[1 mark]

correctly establishes an appropriate expression representing the left-hand side of the inductive step

[1 mark]

uses the assumption within the inductive step

[1 mark]

completes the proof by determining an expression with a factor of 7 representing the right-hand side of the inductive step

[1 mark]

shows logical organisation, having attempted all steps of the proof, including a suitable conclusion

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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