Geometric Progressions

Geometric Progressions: Level 2 Challenges


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?

1+x+x2+x3+x4+=15,x= ?\large 1+x + x^2 + x^3 + x^4 + \ldots = 15 \qquad, \qquad x = \ ?

1+22+322+423++100299=?1+2 \cdot 2+ 3 \cdot 2^2 + 4 \cdot 2^3 + \cdots+ 100 \cdot 2^{99}= \, ?

ab+ab2+ab3+=4 \frac{a}{b}+\frac{a}{b^{2}}+\frac{a}{b^{3}}+\ldots = 4

aa+b+a(a+b)2+a(a+b)3+= ? \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 2+1,1,21, \sqrt2+1,1,\sqrt2-1, find the sum to infinity of all of its terms.

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


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