A sequence $a_1, a_2, a_3, \dots$ is such that $a_n 0$, for \(n \in \mathbb{Z}^+\

Question

A sequence $a_1, a_2, a_3, \dots$ is such that $a_n > 0$, for $n \in \mathbb{Z}^+$, $n \geq 1$. Find $\sum_{n=1}^{N} \ln \left( \frac{a_{n+1}}{a_n} \right)$, and express your answer as a single logarithm.[2] It is also known that this sequence is defined by $\frac{a_{n+1}}{a_n} = \frac{1}{4}$. Given that $\sum_{n=1}^{N} (a_n - a_{n+1})^2 = \frac{k}{16} \sum_{n=1}^{N} (a_n)^2$, find the value of $k$.[2] show that $\sum_{n=1}^{N} (a_n - a_{n+1})^2 = \frac{3}{5} a_1^2 \left( 1 - \frac{1}{16^N} \right)$, for $N \in \mathbb{Z}^+$, $N \geq 1$,[3] find $\sum_{n=2}^{\infty} (a_n - a_{n+1})^2$, leaving your answers in terms of $a_1$.[3]

Accepted Answer

(a) $\ln \left( \frac{{{a}_{N+1}}}{{{a}_{1}}} \right)$ (b) $9$ (ii) $\frac{3}{80}{{a}_{1}}^{2}$

2023 ACJC Promo Q10

2023 ACJC Promo Q10