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Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$.

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Spring Frames

Stage: 5 Challenge Level: Challenge Level:1

Light springs with the same spring constant are attached to light particles and joined to the corners of the following rigid frames, drawn on unit grids (assume that the pentagon and the small triangle are regular):



In each case, where will the particle be at equilibrium and how long will the springs be?

Can you devise any frames for which the equilibrium point would lie outside the frame? On the frame? Can you devise any frames with multiple equilibrium points?

Imagine pulling the particle slightly in each case. In which directions will the particles oscillate to and fro in a straight line? For what sorts of frames would this always be possible?

Extension: If one of the springs has double the spring constant of the other springs in each case, where would the equilibrium points lie?