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?

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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 \(a_n < 0.001\)?

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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 \(k?\)

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\(3\) 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?

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