A sequence $\{a_n\}$ is defined by $a_{1}=2$ and for $n\ge 1$ by the recurrence $a_{n+1}=3a_{n}+4$. (i) Find $a_{5}$. (ii) Find the sum $S_{5}=a_{1}+a_{2}+a_{3}+a_{4}+a_{5}$. (iii) Give a closed‑form expression for the general term $a_{n}$.

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a_5 = 322; \; S_5 = 474; \; a_n = 4\cdot3^{n-1} - 2

Gce Model Questions Vol1 2017 Mathematics Question 40

A sequence \(\{a_n\}\) is defined by \(a_{1}=2\) and for \(n\ge 1\) by the recurrence \(a_{n+1}=3a_{n}+4\). (i) Find \(a_{5}\). (ii) Find the sum \(S_{5}=a_{1}+a_{2}+a_{3}+a_{4}+a_{5}\). (iii) Give a closed‑form expression for the general term \(a_{n}\).

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