| Exam Board | SPS |
|---|
| Module | SPS FM (SPS FM) |
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| Year | 2022 |
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| Session | October |
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| Topic | Proof by induction |
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7. A sequence of positive integers is defined by
$$\begin{aligned}
u _ { 1 } & = 1
u _ { n + 1 } & = u _ { n } + n ( 3 n + 1 ) , \quad n \in \mathbb { Z } ^ { + }
\end{aligned}$$
Prove by induction that
$$u _ { n } = n ^ { 2 } ( n - 1 ) + 1 , \quad n \in \mathbb { Z } ^ { + }$$
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