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I am being asked how to do this proof for an assignment, but am not certain on what is being asked or how to go about doing it.

(a) fix $r\in \mathbb R \setminus \{1\}$. Prove by induction on n that the following statement holds: $$ 1 + r + r^2+…+r^n= \frac{1-r^{n + 1}}{1 - r} \forall r \in \mathbb R$$ (b) Derive the result by setting $S = 1 + r + \cdots + r^n$, multiplying this equation by r, and solving the two equations for S.

insipidintegrator
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Wremmurg
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1 Answers1

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To calculate $1+r+r^2+...r^n$, multiply by $1$:

$$ \frac{1-r}{1-r}\left(1+r+r^2+...r^n\right)=\frac{1}{1-r}\left((1+r+r^2+...r^n)-(r+r^2+...r^n+r^{n+1})\right)=\frac{1-r^{n+1}}{1-r} $$

Logan Post
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  • Before posting an answer to what is likely a duplicate question, please try to search the site for a good target, and flag the question as a duplicate. – Xander Henderson Sep 20 '22 at 15:53