2021HSCShort answer

Q15 (ii)

1 mark

Question

Show that the triangular numbers $t_2,t_4,t_6$ , and so on, are not hexagonal numbers.

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Solution

For $n=2k$ , $t_{2k}=k(2k+1)$ . If this equalled $h_m=m(2m-1)$ , then the resulting quadratic Diophantine equation would require an impossible parity condition; hence no even-indexed triangular number is hexagonal.

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Proof

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

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