If \(x + \frac{1}{x} = 7\), find \(x^5 + \frac{1}{x^5}\).
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 \(x^5 + \frac{1}{x^5} = 15127\).
Important Questions from Algebra
- If $Pe^x = Qe^{-x}$ for all real values of $x$, which one of the following statements is true?
The relationship between two variables $x$ and $y$ is given by $x + py + q = 0$ and is shown in the figure. Find the values of $p$ and $q$.
Note: The figure shown is representative.
The real variables $x, y, z$ and the real constants $p, q, r $ satisfy
$\frac{x}{pq - r^2} = \frac{y}{qr - p^2} = \frac{z}{rp - q^2}$
Given the denominators are non-zero, the value of $px + qy + rz$ isThe complex function
$e^{-\left(\frac{2}{z-1}\right)}$
has __________________Consider two matrices: $P = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$ and $Q = \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}$.
Which of the following statement is/are true?
