An Alternating Coefficient Recursion

★ 8/10Geometric Progression Recursion sequences Telescoping inequality

Problem

Sequences $\{a_n\}$ and $\{b_n\}$ satisfy

b_na_n + a_{n+1} + b_{n+1}a_{n+2} = 0, \qquad b_n = \frac{3 + (-1)^n}{2},

with $a_1 = 2$ and $a_2 = 4$ .

1. Find $a_3, a_4, a_5$ . 2. Let $c_n = a_{2n-1} + a_{2n+1}$ . Prove that $\{c_n\}$ is geometric. 3. Let $S_k = a_2 + a_4 + \cdots + a_{2k}$ . Prove that

\sum_{k=1}^{4n} \frac{S_k}{a_k} < \frac{7}{6} \qquad (n \in \mathbb{N}^*).

Answer

Solution

Difficulty	8/10
Topics	Geometric Progression, Recursion, sequences, Telescoping, inequality

Whiteboard

Your sketch is saved only in this browser. To share it, export your drawing as an image (whiteboard menu → Export as → PNG), then upload that image in the comments below.

Discussion

Ask questions, share alternate solutions, and use LaTeX freely.

0 comments

No comments yet.