To solve the equation [tex]\(\frac{2^n + 2^{n-1}}{2^{n+1} - 2^n} = \frac{3}{2}\)[/tex], let's work through it step-by-step.
1. Simplify the Exponents in the Denominator:
First, recognize that [tex]\(2^{n+1} = 2^n \cdot 2\)[/tex]. Therefore:
[tex]\[
2^{n+1} - 2^n = 2^n \cdot 2 - 2^n = 2 \cdot 2^n - 2^n = 2^n (2 - 1) = 2^n
\][/tex]
2. Simplify the Exponents in the Numerator:
Next, observe that [tex]\(2^{n-1} = \frac{2^n}{2}\)[/tex]. Thus:
[tex]\[
2^n + 2^{n-1} = 2^n + \frac{2^n}{2} = 2^n + \frac{2^n}{2} = 2^n + \frac{2^n}{2} = 2^n \left(1 + \frac{1}{2}\right) = 2^n \cdot \frac{3}{2}
\][/tex]
3. Substitute the Simplified Expressions into the Equation:
Now, substitute the simplified expressions back into the original equation:
[tex]\[
\frac{2^n \cdot \frac{3}{2}}{2^n} = \frac{3}{2}
\][/tex]
By simplifying the fraction:
[tex]\[
\frac{2^n \cdot \frac{3}{2}}{2^n} = \frac{3}{2}
\][/tex]
Which simplifies to:
[tex]\[
\frac{3}{2} = \frac{3}{2}
\][/tex]
4. Analyze the Equation:
What we see here is that after simplifying both sides, the equation reduces to [tex]\(\frac{3}{2} = \frac{3}{2}\)[/tex].
5. Conclusion:
This result is always true and does not depend on [tex]\(n\)[/tex]. Thus, there is no specific value of [tex]\(n\)[/tex] that exclusively satisfies this equation because it is true for all [tex]\(n\)[/tex]. Therefore, there are no specific solutions or unique values for [tex]\(n\)[/tex].
Thus, the conclusion is:
[tex]\[
\boxed{}
\][/tex]
indicating that there is no specific solution for [tex]\(n\)[/tex].
