A sequence is defined by $T_{n+2} = T_{n+1} + T_n$ with $T_1 = 3$, $T_2 = 1$ for \(n \ge

Question

A sequence is defined by $T_{n+2} = T_{n+1} + T_n$ with $T_1 = 3$, $T_2 = 1$ for $n \ge 1$. Another sequence is defined by $r_n = \frac{T_{n+1}}{T_n}$ for $n \ge 1$. Find $T_4$.[1] Show that $r_{n+1} = 1 + \frac{1}{r_n}$ for all for $n \ge 1$.[2] It is given that $r_n$ converges to a limit $L$. Find the exact value of $L$.[3] A third sequence $u_n$, for $n \ge 1$, is an arithmetic sequence with first term $5$ and common difference $2$. Find the least value of $n$ such that $\displaystyle \sum_{k=1}^n u_k > 930$.[2]

Accepted Answer

(a) $T_4=5$ (b) $r_{n+1}=1+\frac1{r_n}$ (c) $L=\frac{1+\sqrt5}{2}$ (d) $n=29$

2025 MI P2 Q4

2025 MI P2 Q4

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