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# Geometric Progressions

A geometric progression is a sequence of numbers where the previous term is multiplied by a constant to get the next term. 1, 2, 4, 8,... is a geometric sequence where each term is multiplied by 2.

# Geometric Progressions: Level 3 Challenges

Given $$p$$ arithmetic progressions, each of which consisting of $$n$$ terms, if their first terms are $$1,2,3,\ldots,p-1,p$$ and common differences are $$1,3,5,7,\ldots,2p-3,2p-1,$$ respectively, what is the sum of all the terms of all the arithmetic progressions?

$(\log_{b} 2)^{0}\left(\log_{b} 5^{2^{0}}\right) + (\log_{b} 2)^{1}\left(\log_{b} 5^{2^{1}}\right) + (\log_{b} 2)^{2}\left(\log_{b} 5^{2^{2}}\right) + \cdots$

If $$b = 100,$$ what is the infinite sum above?

$\{ a + \log_2 3, a + \log_4 3, a + \log_8 3 \}$

Let $$a$$ be a real number such that the above set of numbers form a geometric progression (in that order). Find the common ratio of this geometric progression.

$\Large \sqrt [4]{2}\times \sqrt[8]{4}\times \sqrt[16]{8}\times \sqrt[32]{16}\times \cdots = \ ?$

$1= x^{2} - x^{3} + x^{4} -x^{5} +\ldots$Solve for $$x$$ in the equation above.
If the sum of all values of $$x$$ can be represented in the form $$\dfrac{a+b\sqrt{c}}{d}$$, such that $$a,b,c$$ and $$d$$ are integers and the fraction is in lowest form and $$d> 0$$, find $$a+b+c+d$$.