Higher January 2019 Paper 1R Q12
12 Here are the first four terms of a sequence of fractions.
\[\frac{1}{1} \qquad \frac{2}{3} \qquad \frac{3}{5} \qquad \frac{4}{7}\]The numerators of the fractions form the sequence of whole numbers 1 2 3 4 …
The denominators of the fractions form the sequence of odd numbers 1 3 5 7 …
(a) Write down an expression, in terms of \(n\), for the \(n\)th term of this sequence of fractions. (2)
(b) Using algebra, prove that when the square of any odd number is divided by 4 the remainder is 1 (3)
| Scheme | Marks |
|---|---|
| M1 | |
| \(\dfrac{n}{2n - 1}\) | A1 |
| (2) |
Notes
M1: for \(2n \pm k\) oe as the denominator
A1: oe
| Scheme | Marks |
|---|---|
| \((2n - 1)^2 = 4n^2 - 4n + 1\) | M1 |
| \(4(n^2 - n) + 1\) or \(\dfrac{4n^2 - 4n + 1}{4} = n^2 - n + \dfrac{1}{4}\) | M1 |
| Proved | A1 |
| (3) | |
| (5 marks) |
Notes
M1: or \((2n + 1)^2 = 4n^2 + 4n + 1\) ft on \(2n \pm k\) (\(k\) non zero)
M1: or \(4(n^2 + n) + 1\) or \(\dfrac{4n^2 + 4n + 1}{4} = n^2 + n + \dfrac{1}{4}\)
A1: Conclusion