| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2016 |
| Session | June |
| Marks | 4 |
| Topic | Sequences and Series |
| Type | Sum of Powers Using Standard Formulae |
| Difficulty | Moderate -0.5 This is a straightforward application of standard summation formulae (∑r³ and ∑r) requiring algebraic manipulation to reach the given form. While it involves some factorization skill, it's a routine exercise testing recall and algebraic technique rather than problem-solving or insight, making it slightly easier than average. |
| Spec | 4.06a Summation formulae: sum of r, r^2, r^3 |
**Question 1** [4 marks]
$\sum_{r=1}^{n}(8r^3+r) \equiv 8\sum_{r=1}^{n}r^3 + \sum_{r=1}^{n}r$ M1 (Splitting into separate series)
$\equiv 8 \times \frac{1}{4}n^2(n+1)^2 + \frac{1}{2}n(n+1)$ M1 (Both used), M1 (good factorisation attempt)
$\equiv \frac{1}{2}n(n+1)\{4n^2+4n+1\}$
$\equiv \frac{1}{2}n(n+1)(2n+1)^2$ A1 (Legitimate **(AG)**)
**Total: [4]**1 Using standard summation results, show that $\sum _ { r = 1 } ^ { n } \left( 8 r ^ { 3 } + r \right) \equiv \frac { 1 } { 2 } n ( n + 1 ) ( 2 n + 1 ) ^ { 2 }$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2016 Q1 [4]}}