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What's the sum of the first 100 positive integers? How about the first 1000?

\[\large \frac ab \ , \ ab \ , \ a -b \ , \ a+b \]

Above shows real numbers that belong to an arithmetic progression in order. Find the next term of this sequence.

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\[{2^x + 2^{x+1} + 2^{x+2} + \ldots + 2^{x+2015} = 4^x + 4^{x+1} + 4^{x+2} + \ldots + 4^{x+2015}}\]

If \(x\) satisfies the equation above and it can be represented as

\[\large{\log_D \left( \dfrac{A}{1+B^C} \right)}\]

for positive integers \(A,B,C,D\) where \(B\) is prime, then determine the smallest value of \(A + B + C +D\).

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The image above shows a *broken line* (a series of connected line segments) starting at the origin, *O*. The *n*th segment in the broken line has length \(\frac{1}{n}\), and at the end of each segment, the broken line turns \(60^{\circ}\) counter-clockwise.

As the number of segments in the broken line approaches infinity, the final endpoint of the broken line approaches a point *P*. The distance *OP* can be written as \(\frac{a}{b}\pi\), where *a* and *b* are positive coprime integers. Find \(a+b\).

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\[ \cot^{-1} 3 + \cot^{-1} 7 + \cot^{-1} 13 + \cot^{-1} 21 + \cot ^ {-1} 31 + \ldots \]

Evaluate the sum above in degrees.

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