Question
If \(x + \frac{1}{x} = 7\), find \(x^5 + \frac{1}{x^5}\).
The correct answer is
15127
Let \(a_n = x^n + \frac{1}{x^n}\). We are given \(a_1 = 7\), and we use the identity \(a_{n+1} = a_1 \cdot a_n - a_{n-1}\).
First, \(a_2 = a_1^2 - 2 = 49 - 2 = 47\).
Then \(a_3 = a_1 \cdot a_2 - a_1 = 7 \times 47 - 7 = 329 - 7 = 322\).
Then \(a_4 = a_1 \cdot a_3 - a_2 = 7 \times 322 - 47 = 2254 - 47 = 2207\).
Finally, \(a_5 = a_1 \cdot a_4 - a_3 = 7 \times 2207 - 322 = 15449 - 322 = 15127\).
So \(x^5 + \frac{1}{x^5} = 15127\).
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Important Questions from Algebra
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then, the value of $\frac{x}{y} + \frac{y}{x}$ is Given $f(x, y) = x^2 - 2xy + y^2$
The complete contour of the equation $f(x, y) = 1$ is described by the option(s) ___.
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