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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 4

         

If

\[\large 1 + \frac{3}{x} + \frac{5}{x^2} + \frac{7}{x^3} + \frac{9}{x^4} +\ldots = 91,\]

then evaluate

\[\large 1 + \frac{4}{x} + \frac{9}{x^2} + \frac{16}{x^3} + \frac{25}{x^4} +\ldots \]

An ant travels along a straight path with a distance of \(x\) meters. From then on, it turns left and covers \( \frac{2}{3}\) of the straight distance it traveled before turning, and continues doing this until it eventually reaches an unknown point \(P\) (referred to as the red point in the middle of the picture). Refer to the figure above.

The distance of \(P\) from the ant's starting point is \(kx\) meters, where \(k\) is a positive constant which can be expressed in the form \( \frac{a \sqrt {b}}{c} \), where \( \gcd(a,c) = 1\) and \(b\) is square free. Determine the value of \( a + b + c \).

Challenge: Determine an explicit formula for \(k\).

You have a circle (circle 1) inscribed in an equilateral triangle.
Then, you construct another circle (circle 2) that is tangent to two sides of the triangle and to circle 1.
Then, you construct another circle (circle 3) that is tangent to two sides of the triangle and to circle 2.
Then you construct another circle (circle 4) that is tangent to two sides of the triangle and to circle 3.
Then you keep doing this infinitely.
Now, not only do you do this for one angle of the triangle,
but you repeat the same process for the other two angles of the triangle.

By doing the above, you get an infinite number of circles. If the radius of the circle 2 is 1, and if the area covered by all of those circles can be represented in the form \( \dfrac{a \pi}{b} \), where \(a\) and \(b\) are coprime positive integers, find \(a+b\).

\[\large\sum_{k=1}^{\infty} \dfrac{k^2}{2^k} = \, ? \]

\[S=\dfrac{1×2^2}{10}+\dfrac{2×3^2}{10^2}+\dfrac{3×4^2}{10^3}+\dfrac{4×5^2}{10^4}+\cdots\]

If \(S\) is in the form \(\dfrac{A}{B}\), where \(A\) and \(B\) are coprime positive integers, find the value of \(A+B\).

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