-1 + (2!/8) - (3!/27) + (4!/64) - (5!/125)

asked by guest
on Sep 20, 2024 at 8:09 pm



You asked:

Evaluate the expression: \(-1 + \frac{2!}{8} - \frac{3!}{27} + \frac{4!}{64} - \frac{5!}{125}\)

MathBot Answer:

\[-1 + \frac{2!}{8} - \frac{3!}{27} + \frac{4!}{64} - \frac{5!}{125} = - \frac{2803}{1800} = -1.557\overline{2}\]

\[\begin{aligned}-1 + \frac{2!}{8} - \frac{3!}{27} + \frac{4!}{64} - \frac{5!}{125}& = -1 + 2! \cdot \frac{1}{8} - 3! \cdot \frac{1}{27} + 4! \cdot \frac{1}{64} - 5! \cdot \frac{1}{125}\\& = -1 + \frac{2}{8} - 3! \cdot \frac{1}{27} + 4! \cdot \frac{1}{64} - 5! \cdot \frac{1}{125}\\& = -1 + \frac{1}{4} - 3! \cdot \frac{1}{27} + 4! \cdot \frac{1}{64} - 5! \cdot \frac{1}{125}\\& = -1 + \frac{1}{4} - \frac{6}{27} + 4! \cdot \frac{1}{64} - 5! \cdot \frac{1}{125}\\& = -1 + \frac{1}{4} - \frac{2}{9} + 4! \cdot \frac{1}{64} - 5! \cdot \frac{1}{125}\\& = -1 + \frac{1}{4} - \frac{2}{9} + \frac{24}{64} - 5! \cdot \frac{1}{125}\\& = -1 + \frac{1}{4} - \frac{2}{9} + \frac{3}{8} - 5! \cdot \frac{1}{125}\\& = -1 + \frac{1}{4} - \frac{2}{9} + \frac{3}{8} - \frac{120}{125}\\& = -1 + \frac{1}{4} - \frac{2}{9} + \frac{3}{8} - \frac{24}{25}\\& = - \frac{3}{4} - \frac{2}{9} + \frac{3}{8} - \frac{24}{25}\\& = - \frac{35}{36} + \frac{3}{8} - \frac{24}{25}\\& = - \frac{43}{72} - \frac{24}{25}\\& = - \frac{2803}{1800}\end{aligned}\]

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