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.
326 exam-style questions on Edexcel A Level Maths Sequences and Series, covering 3.1 Arithmetic Sequences, 3.2 Arithmetic Series, 3.3 Geometric Sequences, 3.4 Geometric Series, 3.5 Sum to Infinity, 3.6 Sigma Notation, 3.7 Recurrence Relations, and 3.8 Modelling with Series. Each one has a worked solution and a mark scheme showing where the marks go.