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Geometric Progressions

Geometric Progressions: Level 3 Challenges


Let SS denote the sum of an infinite geometric sequence, with S>0S>0. If the second term of this sequence is 1, what is the smallest possible value of SS?

Try Part I.

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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