Standard +0.3 This is a straightforward proof by induction with a recurrence relation. The base case is trivial (u₁ = 2 = 5⁰ + 1), and the inductive step requires only direct substitution of the formula into the recurrence relation and basic algebraic manipulation (5(5^k + 1) - 4 = 5^(k+1) + 1). This is a standard textbook exercise testing routine application of the induction technique with minimal algebraic complexity, making it slightly easier than average.
4. A sequence of numbers is defined by
$$\begin{aligned}
u _ { 1 } & = 2 \\
u _ { n + 1 } & = 5 u _ { n } - 4 , \quad n \geqslant 1 .
\end{aligned}$$
Prove by induction that, for \(n \in \mathbb { Z } ^ { + } , u _ { n } = 5 ^ { n - 1 } + 1\). [0pt]
4. A sequence of numbers is defined by
$$\begin{aligned}
u _ { 1 } & = 2 \\
u _ { n + 1 } & = 5 u _ { n } - 4 , \quad n \geqslant 1 .
\end{aligned}$$
Prove by induction that, for $n \in \mathbb { Z } ^ { + } , u _ { n } = 5 ^ { n - 1 } + 1$.\\[0pt]
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\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q4 [4]}}