Higher June 2019 Paper 2 Q20
20 Expressions for consecutive triangular numbers are
\[\frac{n(n + 1)}{2} \qquad \text{and} \qquad \frac{(n + 1)(n + 2)}{2}\]Prove that the sum of two consecutive triangular numbers is always a square number. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(\dfrac{n^2 + n}{2}\) or \(\dfrac{n^2 + 2n + n + 2}{2}\) or \(\dfrac{n^2 + 3n + 2}{2}\) | M1 | may be seen in stages eg \(n^2 + n\) followed by \(\dfrac{n^2 + n}{2}\) |
| \(\dfrac{n^2 + n}{2}\) and \(\dfrac{n^2 + 2n + n + 2}{2}\) or \(\dfrac{n^2 + n}{2}\) and \(\dfrac{n^2 + 3n + 2}{2}\) | M1dep | may be seen in stages eg \(n^2 + n\) followed by \(\dfrac{n^2 + n}{2}\) and \(n^2 + 3n + 2\) followed by \(\dfrac{n^2 + 3n + 2}{2}\) implies M2 |
| \(\dfrac{2n^2 + 4n + 2}{2}\) or \(n^2 + 2n + 1\) with M2 seen | A1 | oe single fraction with terms collected eg \(\dfrac{4n^2 + 8n + 4}{4}\) |
| \(n^2 + 2n + 1\) and \((n + 1)^2\) with M2A1 seen | A1 | allow \((n + 1)(n + 1)\) for \((n + 1)^2\) |
| Alternative method 2 | ||
| \(\dfrac{n + 1}{2}(n + n + 2)\) | M1 | oe eg \((n + 1)\left(\dfrac{n}{2} + \dfrac{n + 2}{2}\right)\) |
| \(\dfrac{n + 1}{2}(2n + 2)\) or \(\dfrac{n^2 + n}{2} + \dfrac{n^2 + n}{2} + \dfrac{2n + 2}{2}\) with M1 seen | M1dep | |
| \(\dfrac{2n^2 + 4n + 2}{2}\) or \(n^2 + 2n + 1\) with M2 seen | A1 | oe single fraction with terms collected eg \(\dfrac{4n^2 + 8n + 4}{4}\) |
| \(n^2 + 2n + 1\) and \((n + 1)^2\) with M2A1seen | A1 | allow \((n + 1)(n + 1)\) for \((n + 1)^2\) |
| Alternative method 3 | ||
| \(\dfrac{n + 1}{2}(n + n + 2)\) | M1 | oe eg \((n + 1)\left(\dfrac{n}{2} + \dfrac{n + 2}{2}\right)\) |
| \(\dfrac{n + 1}{2}(2n + 2)\) with M1 seen | M1dep | oe eg \(\dfrac{(n + 1)(2n + 2)}{2}\) |
| \((n + 1)^2\) with M2 seen | A2 | A1 \(2(n + 1)\,\dfrac{n + 1}{2}\) or \(\dfrac{2(n + 1)^2}{2}\) allow \((n + 1)(n + 1)\) for \((n + 1)^2\) |
Additional guidance
| Only substituting in values of \(n\) | M0M0A0A0 |
| Consistently using a different letter to \(n\) can score up to M1M1A1A1 | |
| Using two different letters consistently within the two fractions (eg \(n\) replaced by \(x\) in the first equation and \(n\) replaced by \(y\) in the second equation) can score a maximum of M1M1A0A0 unless recovered to the same letter | |
| Multiplying fractions instead of adding can score a maximum of M2A0 | |
| For M marks condone eg \(n2\) for \(2n\) etc | |
| \(n^2 + n\,/2\) and \(n^2 + 3n + 2/2\) recovered to \(\dfrac{2n^2 + 4n + 2}{2}\) and/or \(n^2 + 2n + 1\) and/or \((n + 1)^2\) | M1M1A0A0 |
| \(n^2 + n\,/2\) and \(n^2 + 3n + 2/2\) not recovered | M0M0A0A0 |
| \(n^2 + n\) and \(n^2 + 3n + 2\) recovered to \(\dfrac{2n^2 + 4n + 2}{2}\) and/or \(n^2 + 2n + 1\) and/or \((n + 1)^2\) | M1M1A0A0 |
| \(n^2 + n\) and \(n^2 + 3n + 2\) not recovered | M0M0A0A0 |
| Equating to \(n^2\) in working can score a maximum of M1M1A0A0 (equating to eg \(x^2\) can score up to M1M1A1A1) | |
| \(1n\) is allowed for \(n\) throughout | |
| Alts 2 and 3 \(\dfrac{n + 1}{2}(2n + 2)\) with M1 seen scores M2 If they attempt to expand \((n + 1)(2n + 2)\) use Alt 2 If they attempt to expand \(\dfrac{1}{2}(2n + 2)\) use Alt 3 |