FP2 June 2018 Q1
1.
(a) Express \(\dfrac{1}{(r + 3)(r + 4)}\) in partial fractions. (1)
(b) Hence, using the method of differences, show that \[\sum_{r=1}^{n} \frac{1}{(r + 3)(r + 4)} = \frac{n}{a(n + a)}\] where \(a\) is a constant to be found. (5)
(c) Find the exact value of \(\displaystyle\sum_{r=15}^{30} \frac{1}{(r + 3)(r + 4)}\) (2)
| Scheme | Marks |
|---|---|
| Mark (a) and (b) together – ignore labels | |
| \(\dfrac{1}{(r + 3)(r + 4)} \equiv \dfrac{1}{(r + 3)} - \dfrac{1}{(r + 4)}\) Cao No working needed – ignore any shown | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \(r = 1\): \(\dfrac{1}{4} - \dfrac{1}{5}\) | |
| \(r = 2\): \(\dfrac{1}{5} - \dfrac{1}{6}\) | |
| \(\ldots r = n - 1\): \(\dfrac{1}{(n + 2)} - \dfrac{1}{(n + 3)}\) | |
| \(r = n\): \(\dfrac{1}{(n + 3)} - \dfrac{1}{(n + 4)}\) First 2 and last term or first and last 2 terms required. Must start at \(r = 1\) (First term complete, 2nd and last may be partial or last term complete 1st and penultimate partial.) | M1 |
| \(\displaystyle\sum_{r=1}^{n}\frac{1}{(r + 3)(r + 4)} = \frac{1}{4} - \frac{1}{(n + 4)}\) Cancel terms. | M1A1 |
| \(\displaystyle\sum_{r=1}^{n}\frac{1}{(r + 3)(r + 4)} = \frac{n}{4(n + 4)}\) Find common denominator, dep on second M mark Cso (All M marks required) | dM1 A1cso |
| \((a = 4)\) Need not be shown explicitly | |
| (5) |
Notes
NB: 1 All marks can be awarded if work done with values 1,2,...\(r\) and then \(r\) replaced with \(n\); if no replacement made, deduct final A mark.
2 \(\dfrac{1}{4} - \dfrac{1}{(n + 4)}\) with NO other working gets M0M1A1M1A0 max
(Corrected from the printed mark scheme: the two sums in (b) are printed as \(\displaystyle\sum_{r=1}^{n}\frac{1}{(r + 2)(r + 3)}\); the sum is of \(\dfrac{1}{(r + 3)(r + 4)}\).)
| Scheme | Marks |
|---|---|
| \(\displaystyle\sum_{r=15}^{30}\frac{1}{(r + 3)(r + 4)} = \frac{30}{4(30 + 4)} - \frac{14}{4(14 + 4)}\) Accept \(n = 30\) and \(n = 14\) only in their answer to (b) Must be subtracted | M1 |
| \(= \dfrac{4}{153}\) oe (exact) Exact answer \(\dfrac{4}{153}\) implies method provided no incorrect work seen in (c). | A1 |
| (2) | |
| (8 marks) |
Notes
ALT
| Scheme | Marks |
|---|---|
| Use the method of differences again, starting at \(r = 15\) and ending at \(r = 30\) Complete method | M1 |
| \(= \dfrac{4}{153}\) oe (exact) Correct answer | A1 |