Consider a sequence $(a_n )$ of positive numbers satisfying the given condition

Question

$a_n\cdot a_{n+2} ≤ a_{n+1}^2$. Then $a_n$ is a:

(a) convergent sequence if $a_1≠ 2a_2$

(b) monotonically increasing sequence if $a_1≠ 2a_2$

(c) convergent sequence if $a_1=2a_2$

(d) monotonically increasing sequence if $a_1= 2a_2$

How do I derive a relation between $a_1$ and $a_2$?

Answer 1

Write this as

$$\frac{a_{n+2}}{a_{n+1}} \leq \frac{a_{n+1}}{a_n}$$

The ratio between consecutive terms is always nonincreasing. By the ratio lemma, convergence is guaranteed if the first ratio $a_2/a_1$ is less than $1$.

Moreover, if the first ratio is $\leq 1$, then the sequence must be monotonically decreasing.