Geometric Progressions

Geometric Progressions: Level 3 Challenges


If the lengths of sides AB,AB, BCBC and ACAC in the figure shown form a geometric progression in that order, what is the ratio between ACAC and ABAB to 3 decimal places?

(logb2)0(logb520)+(logb2)1(logb521)+(logb2)2(logb522)+(\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,b = 100, what is the infinite sum above?

{a+log23,a+log43,a+log83} \{ a + \log_2 3, a + \log_4 3, a + \log_8 3 \}

Let aa 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.

24×48×816×1632×= ?\Large \sqrt [4]{2}\times \sqrt[8]{4}\times \sqrt[16]{8}\times \sqrt[32]{16}\times \cdots = \ ?

Give your answer to 3 decimal places.

1=x2x3+x4x5+ 1= x^{2} - x^{3} + x^{4} -x^{5} +\ldots Solve for xx in the equation above.

If the sum of all values of xx can be represented in the form a+bcd\dfrac{a+b\sqrt{c}}{d}, such that a,b,ca,b,c and dd are integers and the fraction is in lowest form and d>0d> 0, find a+b+c+da+b+c+d.


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