| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2022 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Prove recurrence relation formula |
| Difficulty | Standard +0.8 This is a Further Maths question requiring two induction proofs: one involving a recurrence relation with exponential terms requiring algebraic manipulation of powers of 2, and another proving divisibility by 16. Both parts demand careful algebraic handling and are more sophisticated than typical A-level Core proofs, placing it moderately above average difficulty. |
| Spec | 4.01a Mathematical induction: construct proofs |
\begin{enumerate}
\item (i) A sequence of numbers is defined by
\end{enumerate}
$$\begin{gathered}
u _ { 1 } = 3 \\
u _ { n + 1 } = 2 u _ { n } - 2 ^ { n + 1 } \quad n \geqslant 1
\end{gathered}$$
Prove by induction that, for $n \in \mathbb { N }$
$$u _ { n } = 5 \times 2 ^ { n - 1 } - n \times 2 ^ { n }$$
(ii) Prove by induction that, for $n \in \mathbb { N }$
$$f ( n ) = 5 ^ { n + 2 } - 4 n - 9$$
is divisible by 16
\hfill \mbox{\textit{Edexcel F1 2022 Q9 [10]}}