2021HSCShort answer

Q15 (d)

2 marks

Question

Prove that $2^n+3^n\ne5^n$ for all integers $n\ge2$ .

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Solution

Criteria	Marks
Provides correct solution	2
Writes $5^n=(2+3)^n$ , or equivalent merit	1

Sample answer:

$5=2+3$
$5^n=(2+3)^n$
$=2^n+3^n+\binom n1 2\times3^{n-1}+\text{other terms}$
$>2^n+3^n$

$\therefore 2^n+3^n\ne5^n\quad\text{if }n\ge2$

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Syllabus alignment

Proof

The Nature of ProofUse formal proof language, contradiction, counter-examples and rigorous arguments involving numbers and inequalities.

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