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# Sequences and Series

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

# Sequences and Series: Level 4 Challenges

$\large \frac ab \ , \ ab \ , \ a -b \ , \ a+b$

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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