Geometric Sequences Please Help

Question

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546

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+163

The sequence $(a_n)$ is defined by $a_0=2, a_1=1$ and $a_n=a_{n-1}\sqrt{3}-a_{n-2}$
for all $n\geq2$ Find $a_{100}$ .

Thanks in advance.

Firebolt Jul 21, 2020

geno3141 · Answer 1 · 2020-07-21T09:10:56

If you use the rules of the sequence, you will get the first six elements to be unique elements.

When you continue the sequence, you will find that:

a₆ = -a_o a₇ = -a₁ a₈ = - a₂ a₉ = -a₃ a₁₀ = - a₄ a₁₁ = -a₅

Continuing, you find that:

a₁₂ = a₀ a₁₃ = a₁ a₁₄ = a₂ ...

Etc.

Knowing this, can you calculate the value of a₁₀₀?