Standard +0.8 This is a standard proof by induction with divisibility, requiring verification of the base case, assumption step, and inductive step. The algebraic manipulation (factoring out 11 from 2U_n + 9×13×13^n) is straightforward once the structure is recognized. While it requires careful algebraic handling and understanding of induction mechanics, it follows a well-established template without requiring novel insight, making it moderately above average difficulty.
A sequence is defined by \(U_n = 2^{n+1} + 9 \times 13^n\) for positive integer values of \(n\).
Prove by induction that \(U_n\) is divisible by 11.
[5]
A sequence is defined by $U_n = 2^{n+1} + 9 \times 13^n$ for positive integer values of $n$.
Prove by induction that $U_n$ is divisible by 11.
[5]
\hfill \mbox{\textit{SPS SPS FM 2020 Q6 [5]}}