Sure, let's simplify the expression [tex]\(\frac{1}{4}(2^n - 2^{n+2})\)[/tex] step-by-step.
1. Expand the terms inside the parenthesis:
[tex]\[
\frac{1}{4}(2^n - 2^{n+2})
\][/tex]
2. Rewrite the expression [tex]\(2^{n+2}\)[/tex]:
[tex]\[
2^{n+2} = 2^n \cdot 2^2 = 2^n \cdot 4
\][/tex]
3. Substitute [tex]\(2^{n+2}\)[/tex] with [tex]\(2^n \cdot 4\)[/tex]:
[tex]\[
\frac{1}{4}(2^n - 4 \cdot 2^n)
\][/tex]
4. Factor [tex]\(2^n\)[/tex] from the parenthesis:
[tex]\[
\frac{1}{4}(2^n (1 - 4))
\][/tex]
5. Simplify the expression inside the parenthesis:
[tex]\[
1 - 4 = -3
\][/tex]
6. Substitute [tex]\(-3\)[/tex] back into the expression:
[tex]\[
\frac{1}{4}(2^n \cdot (-3))
\][/tex]
7. Multiply the constants [tex]\(\frac{1}{4}\)[/tex] and [tex]\(-3\)[/tex]:
[tex]\[
\frac{1}{4} \cdot (-3) = -\frac{3}{4}
\][/tex]
8. Combine the result with [tex]\(2^n\)[/tex]:
[tex]\[
-\frac{3}{4} \cdot 2^n
\][/tex]
Therefore, the simplified expression is:
[tex]\[
-\frac{3}{4} \cdot 2^n
\][/tex]
Or more compactly:
[tex]\[
-0.75 \cdot 2^n
\][/tex]
