| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and Series |
| Type | Sum of Powers Using Standard Formulae |
| Difficulty | Standard +0.3 This is a straightforward multi-part question combining routine techniques: (a) is simple substitution to verify a root exists, (b)(i) requires standard algebraic manipulation of sum formulae (textbook exercise), (b)(ii) is basic factorization, and (c) applies the derived formula with simple inequality solving. All parts follow predictable patterns with no novel insight required, making it slightly easier than average. |
| Spec | 1.09a Sign change methods: locate roots4.06a Summation formulae: sum of r, r^2, r^34.06b Method of differences: telescoping series |
**(a) Show that the equation $4x^3 - x - 540000 = 0$ has a root, $a$, in the interval $51 < a < 52$. (2 marks)**
**(b) It is given that $S_n = \sum_{r=1}^{n} (2r - 1)^2$.**
(i) Use the formulae for $\sum_{r=1}^{n} r^2$ and $\sum_{r=1}^{n} r$ to show that $S_n = \frac{n}{3}(kn^2 - 1)$, where $k$ is an integer to be found. **(5 marks)**
(ii) Hence show that $6S_n$ can be written as the product of three consecutive integers. **(2 marks)**
**(c) Find the smallest value of $N$ for which the sum of the squares of the first $N$ odd numbers is greater than 180000. (2 marks)**7
\begin{enumerate}[label=(\alph*)]
\item Show that the equation $4 x ^ { 3 } - x - 540000 = 0$ has a root, $\alpha$, in the interval $51 < \alpha < 52$.
\item It is given that $S _ { n } = \sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 }$.
\begin{enumerate}[label=(\roman*)]
\item Use the formulae for $\sum _ { r = 1 } ^ { n } r ^ { 2 }$ and $\sum _ { r = 1 } ^ { n } r$ to show that $S _ { n } = \frac { n } { 3 } \left( k n ^ { 2 } - 1 \right)$, where $k$ is an integer to be found.
\item Hence show that $6 S _ { n }$ can be written as the product of three consecutive integers.
\end{enumerate}\item Find the smallest value of $N$ for which the sum of the squares of the first $N$ odd numbers is greater than 180000 .
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2013 Q7 [11]}}