The sum of the infinite geometric series \( S = 8 + 4 + 2 + \cdots \) is determined by first identifying the first term and the common ratio. The initial term \( a \) is 8.
The common ratio \( r \) is calculated by dividing the second term by the first term:
\( r = \frac{4}{8} = \frac{1}{2} \)
An infinite geometric series converges if the absolute value of the common ratio is less than 1, that is, \( |r| < 1 \). Here, \( |r| = \frac{1}{2} < 1 \), confirming convergence.
The formula for the sum of a convergent infinite geometric series is:
\( S = \frac{a}{1 - r} \)
Substituting the identified values:
\( S = \frac{8}{1 - \frac{1}{2}} \)
The calculation proceeds as follows:
\( S = \frac{8}{\frac{1}{2}} = 8 \times 2 = 16 \)
Therefore, the sum of the given infinite geometric series is \( 16 \).
If \( A = \begin{bmatrix} 1 & 0 \\ 1/2 & 1 \end{bmatrix} \), then \( A^{50} \) is:
The range of the function \( f(x) = \sin^{-1}(x - \sqrt{x}) \) is equal to?
The function \( f(x) = \tan^{-1} (\sin x + \cos x) \) is an increasing function in: