Evaluate the sum of the convergent infinite geometric series $\sum_{n = 1}^{\infty} 12 (\frac{1}{3})^{n}$ .

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Find the value of $k$ such that the polynomial $P (x) = 2 x^{3} - 3 x^{2} + k x - 6$ is divisible by $(x - 2)$ .
Solve the inequality $\frac{x - 3}{x + 2} \leq 0$ for $x \in R$ .
Find the real solution to the equation $4^{x} - 3 \cdot 2^{x} - 4 = 0$ .
Solve the logarithmic equation $lo g_{2} (x + 3) + lo g_{2} (x - 1) = 5$ for $x \in R$ .
Given that $sin (θ) = \frac{3}{5}$ , calculate the exact value of $cos (2 θ)$ .
An arithmetic progression has first term $a_{1} = 4$ and common difference $d = 3$ . Compute the sum of the first 20 terms, $S_{20}$ .