| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2020 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Prove summation with fractions |
| Difficulty | Standard +0.8 This is a Further Maths F1 proof by induction question with two parts. Part (i) requires algebraic manipulation of fractions with squared terms in denominators, which is more complex than standard summation proofs. Part (ii) is a divisibility proof requiring modular arithmetic reasoning. Both are standard induction exercises for Further Maths but require careful algebra and are more demanding than typical A-level Pure questions. |
| Spec | 4.01a Mathematical induction: construct proofs |
| VILU SIHI NI JIIIM ION OC | VIUV SIHI NI III M M I ON OO | VIAV SIHI NI JIIIM I ION OC |
| VIXV SIHIANI III IM IONOO | VIAV SIHI NI JYHAM ION OO | VI4V SIHI NI JLIYM ION OO |
| END |
\begin{enumerate}
\item (i) Prove by induction that, for $n \in \mathbb { Z } ^ { + }$
\end{enumerate}
$$\sum _ { r = 1 } ^ { n } \frac { 2 r ^ { 2 } - 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } } = \frac { n ^ { 2 } } { ( n + 1 ) ^ { 2 } }$$
(ii) Prove by induction that, for $n \in \mathbb { Z } ^ { + }$
$$f ( n ) = 12 ^ { n } + 2 \times 5 ^ { n - 1 }$$
is divisible by 7
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
VILU SIHI NI JIIIM ION OC & VIUV SIHI NI III M M I ON OO & VIAV SIHI NI JIIIM I ION OC \\
\hline
\end{tabular}
\end{center}
\includegraphics[max width=\textwidth, alt={}, center]{a3457c24-fbda-413d-b3b2-6be375307318-29_2255_50_314_34}\\
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
VIXV SIHIANI III IM IONOO & VIAV SIHI NI JYHAM ION OO & VI4V SIHI NI JLIYM ION OO \\
\hline
\end{tabular}
\end{center}
\includegraphics[max width=\textwidth, alt={}, center]{a3457c24-fbda-413d-b3b2-6be375307318-31_2255_50_314_34}\\
\begin{center}
\begin{tabular}{|l|l|}
\hline
\hline
END & \\
\hline
\end{tabular}
\end{center}
\hfill \mbox{\textit{Edexcel F1 2020 Q8 [12]}}