Algebra
# Geometric Progressions

An infinite crowd of mathematicians enters a bar. The first one orders a pint, the second one orders a half pint, the third one orders a quarter pint, and so on.

The bartender interrupts, "I get it!" – then pours the order for all of the mathematicians. How many pints does the bartender pour?

\[\large 1+x + x^2 + x^3 + x^4 + \ldots = 15 \qquad, \qquad x = \ ? \]

\[1+2 \cdot 2+ 3 \cdot 2^2 + 4 \cdot 2^3 + \cdots+ 100 \cdot 2^{99}= \, ?\]

\[ \frac{a}{b}+\frac{a}{b^{2}}+\frac{a}{b^{3}}+\ldots = 4\]

\[ \frac{a}{a+b}+\frac{a}{(a+b)^{2}}+\frac{a}{(a+b)^{3}}+\ldots = \ ? \]

If the first three terms of a geometric progression are given to be \( \sqrt2+1,1,\sqrt2-1, \) find the sum to infinity of all of its terms.

If the answer is in the form of \( \frac{a+b\sqrt c}d \) for positive integers \(a,b,c,\) and \(d\) with \(c\) square-free, find the minimum value of \(a+b+c+d\).

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