Calculate the sum
$6+66+666+6666...+\underbrace{666...6} n6's (n\geq 1)$
"Questions we can't answer are far better than answers we can't question"
Re: Calculate the sum
\[\begin{eqnarray}
6+66+666+\cdot\cdot\cdot+\underbrace{666...666}_{n~6's}&=&\sum_{k=1}^{n}\left[\dfrac{6}{9}\left(10^k-1\right)\right] \\ &=& \dfrac{6}{9}\sum_{k=1}^n \left(10^k-1\right)\\ &=&\dfrac{6}{9}\left(\sum_{k=1}^n 10^k-\sum_{k=1}^n 1\right) \\&=& \dfrac 6 9 \left(\dfrac{10^{n+1}-10}{9}-n\right) \\ &=& \boxed{\dfrac{2}{27}\left(10^{n+1}-9n-10\right)}
\end{eqnarray}\]
6+66+666+\cdot\cdot\cdot+\underbrace{666...666}_{n~6's}&=&\sum_{k=1}^{n}\left[\dfrac{6}{9}\left(10^k-1\right)\right] \\ &=& \dfrac{6}{9}\sum_{k=1}^n \left(10^k-1\right)\\ &=&\dfrac{6}{9}\left(\sum_{k=1}^n 10^k-\sum_{k=1}^n 1\right) \\&=& \dfrac 6 9 \left(\dfrac{10^{n+1}-10}{9}-n\right) \\ &=& \boxed{\dfrac{2}{27}\left(10^{n+1}-9n-10\right)}
\end{eqnarray}\]
- What is the value of the contour integral around Western Europe?
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.