| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2012 |
| Session | June |
| Marks | 4 |
| Topic | Sequences and series, recurrence and convergence |
| Type | Standard summation formulae application |
| Difficulty | Moderate -0.8 This is a straightforward application of standard summation formulae (∑r and ∑r²) provided in the formula booklet. It requires splitting the sum into three parts, applying the given formulae, and simplifying algebraically—purely routine manipulation with no problem-solving or insight needed. Easier than average A-level work. |
| Spec | 4.06a Summation formulae: sum of r, r^2, r^3 |
$\sum_{r=1}^{n}(r^2 - r + 1) = \sum_{r=1}^{n}r^2 - \sum_{r=1}^{n}r + \sum_{r=1}^{n}1$ Splitting summation and use of given results **M1**
$= \frac{1}{6}n(n+1)(2n+1) - \frac{1}{2}n(n+1) + n$ 1st for $\Sigma r^2$; 2nd for $\Sigma r$ & $\Sigma 1 = n$ **B1 B1**
$= \frac{1}{3}n(n^2 + 2)$ legitimately **A1**
**[4]**1 Using any standard results given in the List of Formulae (MF20), show that
$$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - r + 1 \right) = \frac { 1 } { 3 } n \left( n ^ { 2 } + 2 \right)$$
for all positive integers $n$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2012 Q1 [4]}}