1MA0 Higher November 2016 Paper 2 Q24
*24 Prove that, for all positive values of \(n\), \[\frac{(n + 2)^2 - (n + 1)^2}{2n^2 + 3n} = \frac{1}{n}\]
(4)
| Working | Answer | Mark | Notes |
|---|---|---|---|
| \((n^2 + 4n + 4) - (n^2 + 2n + 1)\) \(\dfrac{2n + 3}{2n^2 + 3n}\) \(\dfrac{2n + 3}{n(2n + 3)}\) | Proof | 4 | M1 for correct method to expand \((n + 2)^2\) or \((n + 1)^2\) M1 for correct simplification of numerator M1 for factorisation of \(2n^2 + 3n\) or for clearing the fractions on both sides correctly C1 for complete and correct proof OR M1 for \(\{(n + 2) - (n + 1)\}\{(n + 2) + (n + 1)\}\) M1 for \(1 \times (2n + 3)\) M1 for factorisation of \(2n^2 + 3n\) or for clearing the fractions on both sides correctly C1 for complete and correct proof OR M1 for \(n\{(n + 2)^2 - (n + 1)^2\} = (2n^2 + 3n) \times 1\) M1 for \(n(n + 2)^2 - n(n + 1)^2\) or for correct expansion of \((n + 2)^2 - (n + 1)^2\) M1 for correct expansion of \(n\{(n + 2)^2 - (n + 1)^2\}\) C1 for complete and correct proof (must include statement recognising the equality of LHS and RHS) |