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Sequences and Series

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

Level 4

         

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

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

Find the number of \(6\)-term strictly increasing geometric progressions, such that all terms are positive integers less than \(1000.\)

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

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 \(A + B + C +D\).

The image above shows a broken line (a series of connected line segments) starting at the origin, O. The nth 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 \(a+b\).

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

Evaluate the sum above in degrees.

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