997.18 = 100/(1 + k) + 100/(1 + k)^2 + 1100/(1 + k)^3

asked by guest
on Nov 18, 2024 at 7:30 am



You asked:

Solve the equation \(997.18 = \frac{100}{1 + k} + \frac{100}{{\left( 1 + k \right)}^{2}} + \frac{1100}{{\left( 1 + k \right)}^{3}}\) for the variable \(k\).

MathBot Answer:

The 3 solutions to the equation are: \[\begin{aligned}k &= - \frac{144577}{149577} + \sqrt[3]{\frac{1851529649142500}{3346527942363033} + \frac{2500 \sqrt{2723646160409}}{7457759643}} + \frac{772885000}{22373278929 \sqrt[3]{\frac{1851529649142500}{3346527942363033} + \frac{2500 \sqrt{2723646160409}}{7457759643}}} \approx 0.10113623\\k &= - \frac{144577}{149577} + \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{2500 \sqrt{2723646160409}}{7457759643} + \frac{1851529649142500}{3346527942363033}} + \frac{772885000}{22373278929 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{2500 \sqrt{2723646160409}}{7457759643} + \frac{1851529649142500}{3346527942363033}}} \approx -1.5004267 -0.86681387 i\\k &= - \frac{144577}{149577} + \frac{772885000}{22373278929 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{2500 \sqrt{2723646160409}}{7457759643} + \frac{1851529649142500}{3346527942363033}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{2500 \sqrt{2723646160409}}{7457759643} + \frac{1851529649142500}{3346527942363033}} \approx -1.5004267 + 0.86681387 i\end{aligned}\]

\(i\) is the imaginary unit, defined as \(i^2 = -1\).

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