Algebra

# 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?

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