Consider a sequence $\left\{ a_{n} \right\}$ with $a_{1} = 2$ and $a_{n} = \frac{a_{n - 1}^{2}}{a_{n

Question

Consider a sequence $\left\{ a_{n} \right\}$ with $a_{1} = 2$ and $a_{n} = \frac{a_{n - 1}^{2}}{a_{n - 2}}\forall n \geq 3,n \in N$, terms of the sequence being distinct. Given that $a_{2}$ and $a_{5}$ are positive integers and $a_{5} \leq 200$, then possible value(s) of $a_{5}$ can be

Accepted Answer

Sol. [Key Concept: Distinct-term sequence with integer constraints ⇒ determine the terms.]$${\left\{ a_{n} ight\} = \left\{ 2,2r,2r^{2},2r^{3},2r^{4},\ldots ight\} }{2r,2r^{4} \in I\& 2r^{4} \leq 200 \Rightarrow r^{4} \leq 100 }$$$\Rightarrow \ r = 2$ or 3$$a_{5} = 2(2)^{4},2(3)^{4} = 32,162$$✘ (a) $a_{5} 
eq 8$✔ (b) $r = 2:\ a_{5 = 2} \cdot 2^{4 = 32}$✔ (c) $r = 3:\ a_{5 = 2} \cdot 3^{4 = 162}$✘ (d) $a_{5} 
eq 200$

Question

Options

Solution

More Progression (Sequence and Series) Questions