The power output of a prototype micro-sensor, PnP_nPn (in microwatts), during its nnn-th hour of operation is modeled by the geometric sequence Pn=45(0.8)nP_n = 45(0.8)^nPn=45(0.8)n for n∈Nn \in \mathbb{N}n∈N. Calculate the total energy ∑n=1∞Pn\sum_{n=1}^{\infty} P_n∑n=1∞Pn consumed by the sensor if it operates indefinitely.
A sequence of experimental index values v1,v2,v3,… v_1, v_2, v_3, \dots\,v1,v2,v3,… is defined by the recurrence relation:
vn+1=11−vn,n∈N v_{n+1} = \frac{1}{1 - v_n}, \quad n \in \mathbb{N} vn+1=1−vn1,n∈NGiven that the initial value is v1=23\displaystyle v_1 = \frac{2}{3}v1=32:
Show that this sequence is periodic.
State the order of this sequence.
Determine the sum of the first 100 terms of the sequence, ∑n=1100vn\sum_{n=1}^{100} v_n∑n=1100vn.
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.