নিচের সমীকরণটার সব সমাধানের যোগফল বের করো।
\[5^{2r+1}+5^2=5^r+5^{r+3}\]
Find the sum of all solutions of the equation, \[5^{2r+1}+5^2=5^r+5^{r+3}\]
BdMO National 2021 Junior Problem 2
- Anindya Biswas
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"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
— John von Neumann
— John von Neumann
- gwimmy(abid)
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Re: BdMO National 2021 Junior Problem 2
$\begin{align*} &5^{2r+1}+5^2 = 5^r+5^{r+3} \\
\Longrightarrow & 5 \cdot (5^r)^2 + 25 = 5^r + 125 \cdot 5^r\\
\Longrightarrow & 5 \cdot (5^r)^2 + 25 = 126\cdot 5^r \\
\Longrightarrow & 5 \cdot (5^r)^2 - 126\cdot 5^r + 25 = 0 \\
\Longrightarrow & 5 \cdot (5^r)^2 -125\cdot 5^r -5^r + 25 = 0\\
\Longrightarrow & 5 \cdot (5^r) (5^r - 25) - 1 (5^r -25)=0\\
\Longrightarrow & (5^r - 25)(5^r - 1) =0 \\
\Longrightarrow & 5^r = 25\text{ or }5^r = \frac{1}{5}\\
\Longrightarrow & r = \boxed{2} \text{ or } r = \boxed{-1}
\end{align*}$
\Longrightarrow & 5 \cdot (5^r)^2 + 25 = 5^r + 125 \cdot 5^r\\
\Longrightarrow & 5 \cdot (5^r)^2 + 25 = 126\cdot 5^r \\
\Longrightarrow & 5 \cdot (5^r)^2 - 126\cdot 5^r + 25 = 0 \\
\Longrightarrow & 5 \cdot (5^r)^2 -125\cdot 5^r -5^r + 25 = 0\\
\Longrightarrow & 5 \cdot (5^r) (5^r - 25) - 1 (5^r -25)=0\\
\Longrightarrow & (5^r - 25)(5^r - 1) =0 \\
\Longrightarrow & 5^r = 25\text{ or }5^r = \frac{1}{5}\\
\Longrightarrow & r = \boxed{2} \text{ or } r = \boxed{-1}
\end{align*}$
Umm....the healer needs healing...
Re: BdMO National 2021 Junior Problem 2
My solution
- gwimmy(abid)
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- Joined:Tue Apr 06, 2021 11:23 am
Re: BdMO National 2021 Junior Problem 2
Umm....the healer needs healing...