| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2017 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Prove summation formula |
| Difficulty | Standard +0.3 This is a standard Further Maths proof by induction question with two routine parts: (i) proving a summation formula using standard algebraic manipulation, and (ii) proving divisibility which requires factoring out the common term. Both follow textbook templates with no novel insights required, making it slightly easier than average even for Further Maths. |
| Spec | 4.01a Mathematical induction: construct proofs |
\begin{enumerate}
\item (i) Prove by induction that, for $n \in \mathbb { Z } ^ { + }$
\end{enumerate}
$$\sum _ { r = 1 } ^ { n } \left( 4 r ^ { 3 } - 3 r ^ { 2 } + r \right) = n ^ { 3 } ( n + 1 )$$
(ii) Prove by induction that, for $n \in \mathbb { Z } ^ { + }$
$$f ( n ) = 5 ^ { 2 n } + 3 n - 1$$
is divisible by 9
\hfill \mbox{\textit{Edexcel F1 2017 Q9 [12]}}