Higher June 2022 Paper 1 Q23
23 A polygon has \(n\) sides, where \(n \gt 5\)
When arranged in order of size, starting with the largest number, the sizes of the interior angles of the polygon, in degrees, are the terms of an arithmetic sequence.
Here are the first five terms of this sequence.
177 175 173 171 169
Find the value of \(n\)
Show clear algebraic working.
(6)
| Scheme | Marks |
|---|---|
| \(d = -2\) | M1 |
\((S_n =)\;\dfrac{n}{2}\left[2(177) + (n - 1)(-2)\right]\) or \((S_n =)\;\dfrac{n}{2}\left[354 - 2n + 2\right]\) or \((S_n =)\;\dfrac{n}{2}\left[356 - 2n\right]\) oe | M1 |
| \(\dfrac{n}{2}\left[2(177) + (n - 1)(-2)\right] = (n - 2) \times 180\) | M1 |
| E.g. \(2n^2 + 4n - 720 = 0\) or \(n^2 + 2n - 360 = 0\) oe Allow \(n^2 + 2n = 360\) | A1 |
E.g. \((x - 18)(x + 20)\;(= 0)\) \(x = \dfrac{-2 \pm \sqrt{2^2 - 4 \times 1 \times -360}}{2}\) e.g. \((x + 1)^2 - (1)^2 = 360\) | M1 |
| Working required Answer: 18 | A1 |
| (6) | |
| (6 marks) |
Notes
M1: for common difference
M1: dep on M2 for equating \(S_n\) with \((n - 2) \times 180\)
A1: (dep on M3) writing a correct 3-term quadratic expression in form \(ax^2 + bx + c\;(= 0)\)
allow \(ax^2 + bx = c\)
A1: dep on M3 for 18 only
| Scheme | Marks |
|---|---|
| 3, 5, 7,… and \(d = 2\) or \(a = 3\) and \(d = 2\) | M1 |
\((S_n =)\;\dfrac{n}{2}\left[2(3) + (n - 1)(2)\right]\) or \((S_n =)\;\dfrac{n}{2}\left[6 + 2n - 2\right]\) or \((S_n =)\;\dfrac{n}{2}\left[4 + 2n\right]\) oe | M1 |
| \(\dfrac{n}{2}\left[2(3) + (n - 1)(2)\right] = 360\) | M1 |
| E.g. \(2n^2 + 4n - 720 = 0\) or \(n^2 + 2n - 360 = 0\) oe Allow \(n^2 + 2n = 360\) | A1 |
E.g. \((x - 18)(x + 20)\;(= 0)\) \(x = \dfrac{-2 \pm \sqrt{2^2 - 4 \times 1 \times -360}}{2}\) e.g. \((x + 1)^2 - (1)^2 = 360\) | M1 |
| Working required Answer: 18 | A1 |
Notes
M1: for identifying exterior angle sequence for at least 3 terms and \(d = 2\) or first term and common difference
M1: for correctly substituting 3 and 2 into \((S_n =)\;\dfrac{n}{2}\left[2a + (n - 1)d\right]\)
M1: dep on M2 for equating \(S_n\) with 360
A1: (dep on M3) writing a correct 3-term quadratic expression in form \(ax^2 + bx + c\;(= 0)\)
allow \(ax^2 + bx = c\)
A1: dep on M3 for 18 only