A Cosine Value Set

★ 6/10trigonometry Geometric Progression Recursion sequences Trigonometric Identities

Problem

The sequence $\{a_n\}$ has partial sums $S_n$ , with $a_1 = \dfrac{1}{2}$ and

\frac{S_n}{a_n} = 2^n - \lambda \quad \text{for all } n \in \mathbb{N}^*,

where $\lambda$ is a constant.

1. Find $\lambda$ and a closed formula for $a_n$ . 2. Find the number of elements of the set

A = \left\{x \;\middle|\; x = \cos\frac{\pi}{12 a_n},\ n \in \mathbb{N}^*\right\}.

Answer

Solution

Difficulty	6/10
Topics	trigonometry, Geometric Progression, Recursion, sequences, Trigonometric Identities

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