Let's solve the equation step-by-step:
[tex]\[ 5^{n-1} + \frac{5}{5^n} = 5 \frac{1}{5} \][/tex]
First, convert the mixed number [tex]\(5 \frac{1}{5}\)[/tex] to an improper fraction:
[tex]\[ 5 \frac{1}{5} = 5 + \frac{1}{5} = \frac{5 \cdot 5}{5} + \frac{1}{5} = \frac{25}{5} + \frac{1}{5} = \frac{26}{5} \][/tex]
So the equation becomes:
[tex]\[ 5^{n-1} + \frac{5}{5^n} = \frac{26}{5} \][/tex]
Next, let's simplify [tex]\(\frac{5}{5^n}\)[/tex]. We know that:
[tex]\[ \frac{5}{5^n} = 5 \cdot \frac{1}{5^n} = 5 \cdot 5^{-n} = 5^{1-n} \][/tex]
Therefore, the equation can be written as:
[tex]\[ 5^{n-1} + 5^{1-n} = \frac{26}{5} \][/tex]
We need to solve for [tex]\( n \)[/tex].
After solving the equation, we find that the values of [tex]\( n \)[/tex] are:
[tex]\[ n = 0 \][/tex]
and
[tex]\[ n = 2 \][/tex]
So, the solutions to the equation [tex]\( 5^{n-1} + \frac{5}{5^n} = 5 \frac{1}{5} \)[/tex] are:
[tex]\[ n = 0 \quad \text{and} \quad n = 2 \][/tex]
