A recurrence relation is defined for $n\ge 1$ by \({{t}_{n+1}}=k{{t}_{n}}-1,\quad {{t}_{1}}=2

Question

A recurrence relation is defined for $n\ge 1$ by ${{t}_{n+1}}=k{{t}_{n}}-1,\quad {{t}_{1}}=2$, where $k$ is a non-zero constant. Find the value of ${{t}_{3}}$ in terms of $k$. Given that ${{t}_{3}}=14$ find the possible values of $k$.

Accepted Answer

(a) $t_3=2k^2-k-1$. (b) $k=3$ or $k=-\frac52$.

T.Madas - Recurrence Relations Q8

T.Madas - Recurrence Relations Q8

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