1,230(1+r)^25=8,980

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on Jan 09, 2025 at 10:34 am



You asked:

Solve the equation 1,230(1+r)25=8,9801,230 {\left( 1 + r \right)}^{25} = 8,980 for the variable rr.

MathBot Answer:

The solutions are given by the formula: r=(−1+1232425⋅89825cos⁡(2πk25)123)+1232425⋅89825sin⁡(2πk25)123ir = \left(-1 + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} \cos{\left(\frac{2 \pi k}{25} \right)}}{123}\right) + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} \sin{\left(\frac{2 \pi k}{25} \right)}}{123} i where kk is an integer in 0≤k<250 \le k < 25.

There is 11 real solution.

There are 2424 solutions with nonzero imaginary part.

Real solutions: r=−1+1232425⋅89825123≈0.082766596\begin{aligned}r &= -1 + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898}}{123} \\&\approx 0.082766596\end{aligned}

Solutions with nonzero imaginary part (99 of 2424 displayed): r=−1+1232425⋅89825cos⁡(2π25)123+1232425⋅89825isin⁡(2π25)123≈0.048749492+0.2692731ir=−1+1232425⋅89825cos⁡(4π25)123+1232425⋅89825isin⁡(4π25)123≈−0.051164399+0.52162679ir=−1+1232425⋅89825cos⁡(6π25)123+1232425⋅89825isin⁡(6π25)123≈−0.21069712+0.74120474ir=−1+1232425⋅89825cos⁡(8π25)123+1232425⋅89825isin⁡(8π25)123≈−0.41982465+0.91421007ir=−1+1232425⋅89825(−14+54)123+1232425⋅89825i58+58123≈−0.66540672+1.0297722ir=−1+1232425⋅89825cos⁡(12π25)123+1232425⋅89825isin⁡(12π25)123≈−0.93201252+1.08063ir=−1−1232425⋅89825cos⁡(11π25)123+1232425⋅89825isin⁡(11π25)123≈−1.2028902+1.0635878ir=−1−1232425⋅89825cos⁡(9π25)123+1232425⋅89825isin⁡(9π25)123≈−1.4610196+0.97971651ir=−1−1232425⋅89825cos⁡(7π25)123+1232425⋅89825isin⁡(7π25)123≈−1.6901814+0.834286i\begin{aligned}r &= -1 + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} \cos{\left(\frac{2 \pi}{25} \right)}}{123} + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} i \sin{\left(\frac{2 \pi}{25} \right)}}{123} \\&\approx 0.048749492 + 0.2692731 i\\r &= -1 + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} \cos{\left(\frac{4 \pi}{25} \right)}}{123} + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} i \sin{\left(\frac{4 \pi}{25} \right)}}{123} \\&\approx -0.051164399 + 0.52162679 i\\r &= -1 + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} \cos{\left(\frac{6 \pi}{25} \right)}}{123} + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} i \sin{\left(\frac{6 \pi}{25} \right)}}{123} \\&\approx -0.21069712 + 0.74120474 i\\r &= -1 + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} \cos{\left(\frac{8 \pi}{25} \right)}}{123} + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} i \sin{\left(\frac{8 \pi}{25} \right)}}{123} \\&\approx -0.41982465 + 0.91421007 i\\r &= -1 + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} \left(- \frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{123} + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{123} \\&\approx -0.66540672 + 1.0297722 i\\r &= -1 + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} \cos{\left(\frac{12 \pi}{25} \right)}}{123} + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} i \sin{\left(\frac{12 \pi}{25} \right)}}{123} \\&\approx -0.93201252 + 1.08063 i\\r &= -1 - \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} \cos{\left(\frac{11 \pi}{25} \right)}}{123} + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} i \sin{\left(\frac{11 \pi}{25} \right)}}{123} \\&\approx -1.2028902 + 1.0635878 i\\r &= -1 - \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} \cos{\left(\frac{9 \pi}{25} \right)}}{123} + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} i \sin{\left(\frac{9 \pi}{25} \right)}}{123} \\&\approx -1.4610196 + 0.97971651 i\\r &= -1 - \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} \cos{\left(\frac{7 \pi}{25} \right)}}{123} + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} i \sin{\left(\frac{7 \pi}{25} \right)}}{123} \\&\approx -1.6901814 + 0.834286 i\end{aligned}

ii is the imaginary unit, defined as i2=−1i^2 = -1.

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