 Algebra

# Geometric Progressions: Level 3 Challenges

If the lengths of sides $AB,$ $BC$ and $AC$ in the figure shown form a geometric progression in that order, what is the ratio between $AC$ and $AB$ to 3 decimal places? $(\log_{b} 2)^{0}\left(\log_{b} 5^{2^{0}}\right) + (\log_{b} 2)^{1}\left(\log_{b} 5^{2^{1}}\right) + (\log_{b} 2)^{2}\left(\log_{b} 5^{2^{2}}\right) + \cdots$

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.

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

Give your answer to 3 decimal places.

$1= x^{2} - x^{3} + x^{4} -x^{5} +\ldots$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 $a+b+c+d$.

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