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.
