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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.

Level 3

         

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.

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

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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