The fourth, tenth, and thirteenth terms of a geometric sequence form an arithmetic sequence. Given that the geometric sequence has a sum to infinity, find its' common ratio correct to 3 significant figures.
Geometric and Arithmetic Sequences Q About Common Ratios
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sequences-and-series
1 Answers
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Hint:
We have that if $a_n$ is the $n$th term of the geometric progression, then, $$2ar^9 = ar^3 + ar^{12}$$ $$\Rightarrow r^9 -2r^6 + 1 = 0 \tag{1}$$ assuming $r^3\neq 0$. Let $r^3 = k$, then $(1)$ becomes, $$k^3-2k^2+1=0$$ $$\Rightarrow (k-1)(k^2-k-1) = 0$$ $$\Rightarrow k = r^3 = 1 \text{ or } \frac{1\pm \sqrt{5}}{2}$$ and I hope you can take it from here.