A damped mechanical oscillator's displacement at the peak of each oscillation forms a geometric sequence. The displacement of the kkk-th peak is given by dk=ark−1d_k = ar^{k-1}dk=ark−1.
Prove that the sum of the displacements for the first n n\,n peaks is given by
Sn=a(1−rn)1−r S_n = \frac{a(1 - r^n)}{1 - r} Sn=1−ra(1−rn)The displacement at the 4th peak is 40 mm and the displacement at the 7th peak is -5 mm.
Determine the value of the common ratio rrr.
Hence calculate the sum of the displacements for the first 10 peaks, giving your answer to 2 decimal places.
