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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 2

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?

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

$1+2 \cdot 2+ 3 \cdot 2^2 + 4 \cdot 2^3 + \ldots+ 100 \cdot 2^{99}= \ ?$

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

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

If the first three terms of a geometric progression is 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 $$\dfrac{a+b\sqrt c}d$$ 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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