A sequence of real numbers $x_1$, $x_2$, $x_3$, $\dots$ satisfies the recurrence relation \[

Question

A sequence of real numbers $x_1$, $x_2$, $x_3$, $\dots$ satisfies the recurrence relation $$x_{n+1} = \frac{5x_n + 2}{2x_n + 3} \text{ for all}\hspace{0.5em} n \ge 1.$$ Given that the sequence converges to $l$, find the possible exact values of $l$.[3] Describe how the sequence behaves when $x_1 = 3$.[1] Given that $x_5 = \frac{3503}{2158}$, find the value of $x_1$.[3]

Accepted Answer

(a) $l=\frac{1+\sqrt5}{2}$ or $l=\frac{1-\sqrt5}{2}$ (b) Decreasing and converging to $\frac{1+\sqrt5}{2}$ (c) $x_1=\frac{29}{10}=2.9$

2025 DHS P1 Q4

2025 DHS P1 Q4

Video Solution:

Solution:

Solution locked