(1-(1+r)^-4) = 3.6667 r
You asked:
MathBot Answer:
The solutions are given by the formula: \[r = \left(-1 + \frac{10 \sqrt{3} \cdot 2963^{\frac{3}{4}} \cos{\left(\frac{\pi}{4} + \frac{\pi k}{2} \right)}}{8889}\right) + \frac{10 \sqrt{3} \cdot 2963^{\frac{3}{4}} \sin{\left(\frac{\pi}{4} + \frac{\pi k}{2} \right)}}{8889} i\] where \(k\) is an integer in \(0 \le k < 4\).
There are \(0\) real solutions.
There are \(4\) solutions with nonzero imaginary part.
The are no real solutions.
Solutions with nonzero imaginary part: \[\begin{aligned}r &= -1 + \frac{5 \cdot 2963^{\frac{3}{4}} \sqrt{6}}{8889} + \frac{5 \cdot 2963^{\frac{3}{4}} \sqrt{6} i}{8889} \\&\approx -0.44666077 + 0.55333923 i\\r &= -1 - \frac{5 \cdot 2963^{\frac{3}{4}} \sqrt{6}}{8889} + \frac{5 \cdot 2963^{\frac{3}{4}} \sqrt{6} i}{8889} \\&\approx -1.5533392 + 0.55333923 i\\r &= -1 - \frac{5 \cdot 2963^{\frac{3}{4}} \sqrt{6}}{8889} - \frac{5 \cdot 2963^{\frac{3}{4}} \sqrt{6} i}{8889} \\&\approx -1.5533392 -0.55333923 i\\r &= -1 + \frac{5 \cdot 2963^{\frac{3}{4}} \sqrt{6}}{8889} - \frac{5 \cdot 2963^{\frac{3}{4}} \sqrt{6} i}{8889} \\&\approx -0.44666077 -0.55333923 i\end{aligned}\]
\(i\) is the imaginary unit, defined as \(i^2 = -1\).