Arithmetic and Geometric Progressions Problem Solving

Let \(\{a_n\}\) be a geometric progression such that \(a_1+a_3=15\) and \(a_2+a_4=30\). What is the sum of the first \(5\) terms of this geometric progression?

The sum of three numbers in geometric progression is \(52\). If the product of the three numbers is \(1728\), what is the value of the maximum number?

Let \(\{a_n\}\) be a geometric progression such that
\[a_5=\frac{15}{16} \mbox{ and } a_8=\frac{15}{128}.\]
What is the smallest integer \(n\) for which \(a_n < 0.001\)?

The curve \(y=-x^3+8x^2+40x\) and the line \(y=k\) intersect at \(3\) distinct points. If the \(x\)-coordinates of these \(3\) points form a geometric progression, what is the real number \(k?\)

\(3\) real numbers that form a geometric progression have sum equal to \(175\) and product equal to \(17576\). What is the sum of the largest and smallest numbers?