Algebra
# Geometric Progressions

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

Try Part I.

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

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 $a+b+c+d$.