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Prove by induction that $\displaystyle \sum_{i=0}^{n} (3\cdot 5^i) = {3(5^{n+1}-1) \over 4}$ for all non-negative integers, $n$.

After induction hypothesis my equation becomes

${3(5^{k+1}-1) \over 4} + (3\cdot 5^{k+1}) = {3(5^{k+2}-1) \over 4}$

$\cfrac{15^{k+1}-3}{4} + 15^{k+1} = \cfrac{15^{k+2}-3}{4}$

$\cfrac{75^{k+1}-3}{4} = \cfrac{15^{k+2}-3}{4}$

After that I have no clue what to do.

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    $3 \cdot 5^{k+1} \ne 15^{k+1}\,$ for one thing. – 2017-02-26
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    I am an idiot of highest level. – 2017-02-26
  • 3
    Far from that, real high level idiots never ask a question, nor recognize answers ;-) – 2017-02-26

1 Answers 1

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This is wrongly done. Notice that $3\times5^n\ne(3\times5)^n$. Take $n=0$ for example to see this.

Instead, one should have

$$\begin{align}\frac{3(5^{k+1}-1)}4+3\times5^{k+1}&=\frac{3(5^{k+1}-1+4\times5^{k+1})}4\\&=\frac{3(5\times5^{k+1}-1)}4\\&=\frac{3(5^{k+2}-1)}4\end{align}$$

and we are done!

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    I am such a fool. thank you – 2017-02-26
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    No problem. You welcome :-) – 2017-02-26