Sequences and Series

Sequences and Series: Level 4 Challenges


ab , ab , ab , a+b\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 66-term strictly increasing geometric progressions, such that all terms are positive integers less than 1000.1000.

2x+2x+1+2x+2++2x+2015=4x+4x+1+4x+2++4x+2015{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 xx satisfies the equation above and it can be represented as logD(A1+BC)\log_D \left(\dfrac{A}{1+B^C} \right) for positive integers AA, BB, CC, and DD, where BB is prime, determine the smallest value of A+B+C+DA + 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 1n\frac{1}{n}, and at the end of each segment, the broken line turns 6060^{\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 abπ\frac{a}{b}\pi, where a and b are positive coprime integers. Find a+ba+b.

cot13+cot17+cot113+cot121+cot131+ \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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