3 Consider the sequence $$ \left(u_{n}\right) $$ defined by $$ u_{0}=11 $$, and $$ u_{n+1}=\frac{10}{11} u_{n}+\frac{12}{11}, \forall n \in \mathrm{~N} $$ a) Prove by mathematical induction that $$ u_{n}

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3 Consider the sequence $$ \left(u_{n}\right) $$ defined by $$ u_{0}=11 $$, and $$ u_{n+1}=\frac{10}{11} u_{n}+\frac{12}{11}, \forall n \in \mathrm{~N} $$ a) Prove by mathematical induction that $$ u_{n}<12 $$ for all $$ n \geq 0 $$. b) Show that $$ \left(u_{n}\right) $$ is monotonically increasing. c) Show that $$ \left(u_{n}\right) $$ is convergent. Another sequence $$ \left(v_{n}\right) $$ is defined as $$ v_{n}=u_{n}-12 $$ for all $$ n \geq 0 $$ d) Show that $$ \left(v_{n}\right) $$ is a geometric progression. e) Express $$ v_{n} $$ in terms of n . f) Hence deduce an expression for $$ u_{n} $$ in terms of n and the limit of $$ \left(u_{n}\right) $$. ( 15 marks)

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a) Prove by mathematical induction that $$ u_{n}<12 $$ for all $$ n \geq 0 $$. Base case: $$ u_{0} = 11 < 12 $$. Inductive step: Assume $$ u_{k} < 12 $$, then $$ u_{k+1} = \frac{10}{11} u_{k} + \frac{12}{11} < 12 $$. b) Show that $$ \left(u_{n}\right) $$ is monotonically increasing. $$ u_{n+1} - u_{n} = \frac{12}{11} - \frac{1}{11} u_{n} > 0 $$ since $$ u_{n} < 12 $$. c) Show that $$ \left(u_{n}\right) $$ is convergent. Monotonic and bounded above by 12, $$ \left(u_{n}\right) $$ converges to $$ L = 12 $$. d) Show that $$ \left(v_{n}\right) $$ is a geometric progression. $$ v_{n+1} = \frac{10}{11} v_{n} $$, so $$ \left(v_{n}\right) $$ is a geometric progression with ratio $$ \frac{10}{11} $$. e) Express $$ v_{n} $$ in terms of n. $$ v_{n} = v_{0} \cdot \left(\frac{10}{11}\right)^{n} $$.

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