Higher June 2017 Paper 3 Q22
22 Work out an expression for the \(n\)th term of the quadratic sequence
\[2 \qquad 17 \qquad 40 \qquad 71 \qquad \ldots\]Give your answer in the form \(\quad an^2 + bn + c \quad\) where \(a\), \(b\) and \(c\) are constants. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| Second differences 8 | M1 | Implied by \(4n^2\) |
| Any three values from \(-2 \quad 1 \quad 4 \quad 7\) | M1dep | |
| \(4n^2 + 3n - 5\) | A1 | oe Allow \(\;a = 4 \quad b = 3 \quad c = -5\) |
| Alternative method 2 | ||
| Any 3 of \(a + b + c = 2\) \(4a + 2b + c = 17\) \(9a + 3b + c = 40\) \(16a + 4b + c = 71\) | M1 | Using \(\;an^2 + bn + c\) |
| Any 2 equations in 2 unknowns eg \(3a + b = 15\) \(5a + b = 23\) \(7a + b = 31\) \(8a + 2b = 38\) \(12a + 2b = 54\) \(15a + 3b = 69\) | M1dep | Correctly eliminates the same letter using two different pairs of equations |
| \(4n^2 + 3n - 5\) | A1 | oe Allow \(\;a = 4 \quad b = 3 \quad c = -5\) |
| Alternative method 3 | ||
| Second differences 8 \(a = 4\) or \(c = 2 - 7\;\) or \(-5\) | M1 | Using \(\;an^2 + bn + c\) |
| \(3a + b = 17 - 2\) and substitutes their \(a\) | M1dep | oe \(\;\) eg \(\;b = 3\) May also see \(\;a + b + c = 2\) used to work out \(c\) |
| \(4n^2 + 3n - 5\) | A1 | oe Allow \(\;a = 4 \quad b = 3 \quad c = -5\) |
Additional guidance
Sequence \(\quad (-5) \quad 2 \quad 17 \quad 40 \quad 71\)
1st differences are \(\quad (7) \quad 15 \quad 23 \quad 31\)
2nd differences are \(\quad 8 \quad 8 \quad 8\)