To determine which formula describes the sequence [tex]\(-81, 108, -144, 192, \ldots\)[/tex], we need to evaluate each given formula and compare the sequence it produces to the provided sequence. Let's analyze each formula in turn for the first few terms [tex]\(x = 1, 2, 3, 4\)[/tex].
1. [tex]\(f(x) = -81 \left(\frac{4}{3}\right)^{x-1}\)[/tex]
- For [tex]\(x = 1\)[/tex]: [tex]\( f(1) = -81 \left(\frac{4}{3}\right)^{0} = -81 \)[/tex]
- For [tex]\(x = 2\)[/tex]: [tex]\( f(2) = -81 \left(\frac{4}{3}\right)^{1} = -108 \)[/tex]
- For [tex]\(x = 3\)[/tex]: [tex]\( f(3) = -81 \left(\frac{4}{3}\right)^{2} = -144 \)[/tex]
- For [tex]\(x = 4\)[/tex]: [tex]\( f(4) = -81 \left(\frac{4}{3}\right)^{3} = -192 \)[/tex]
The produced sequence is [tex]\(-81, -108, -144, -192\)[/tex], which does not match.
2. [tex]\(f(x) = -81 \left(-\frac{3}{4}\right)^{x-1}\)[/tex]
- For [tex]\(x = 1\)[/tex]: [tex]\( f(1) = -81 \left(-\frac{3}{4}\right)^{0} = -81 \)[/tex]
- For [tex]\(x = 2\)[/tex]: [tex]\( f(2) = -81 \left(-\frac{3}{4}\right)^{1} = 60.75 \)[/tex]
- For [tex]\(x = 3\)[/tex]: [tex]\( f(3) = -81 \left(-\frac{3}{4}\right)^{2} = -45.5625 \)[/tex]
- For [tex]\(x = 4\)[/tex]: [tex]\( f(4) = -81 \left(-\frac{3}{4}\right)^{3} = 34.171875 \)[/tex]
The produced sequence is [tex]\(-81, 60.75, -45.5625, 34.171875\)[/tex], which does not match.
3. [tex]\(f(x) = -81 \left(-\frac{4}{3}\right)^{x-1}\)[/tex]
- For [tex]\(x = 1\)[/tex]: [tex]\( f(1) = -81 \left(-\frac{4}{3}\right)^{0} = -81 \)[/tex]
- For [tex]\(x = 2\)[/tex]: [tex]\( f(2) = -81 \left(-\frac{4}{3}\right)^{1} = 108 \)[/tex]
- For [tex]\(x = 3\)[/tex]: [tex]\( f(3) = -81 \left(-\frac{4}{3}\right)^{2} = -144 \)[/tex]
- For [tex]\(x = 4\)[/tex]: [tex]\( f(4) = -81 \left(-\frac{4}{3}\right)^{3} = 192 \)[/tex]
The produced sequence is [tex]\(-81, 108, -144, 192\)[/tex], which matches the provided sequence.
4. [tex]\(f(x) = -81 \left(\frac{3}{4}\right)^{x-1}\)[/tex]
- For [tex]\(x = 1\)[/tex]: [tex]\( f(1) = -81 \left(\frac{3}{4}\right)^{0} = -81 \)[/tex]
- For [tex]\(x = 2\)[/tex]: [tex]\( f(2) = -81 \left(\frac{3}{4}\right)^{1} = -60.75 \)[/tex]
- For [tex]\(x = 3\)[/tex]: [tex]\( f(3) = -81 \left(\frac{3}{4}\right)^{2} = -45.5625 \)[/tex]
- For [tex]\(x = 4\)[/tex]: [tex]\( f(4) = -81 \left(\frac{3}{4}\right)^{3} = -34.171875 \)[/tex]
The produced sequence is [tex]\(-81, -60.75, -45.5625, -34.171875\)[/tex], which does not match.
Based on the comparisons, the formula that matches the given sequence [tex]\(-81, 108, -144, 192, \ldots\)[/tex] is
[tex]\[ f(x) = -81 \left(-\frac{4}{3}\right)^{x-1} \][/tex]
