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.
308 exam-style questions on OCR (MEI) A Level Maths 1.6 Sequences and Series, covering 1.6.1 Binomial expansion for positive integer n, 1.6.2 Factorial and combinations notation, 1.6.3 Binomial expansion for rational n (A-level only), 1.6.4 Binomial expansion of (a + bx)^n via factoring (A-level only), 1.6.5 Binomial approximating polynomials (A-level only), 1.6.6 What a sequence is; finite and infinite (A-level only), 1.6.7 Generate a sequence by formula or recurrence (A-level only), 1.6.8 A series as a sum of terms (A-level only), 1.6.9 Sigma notation (A-level only), 1.6.10 Increasing, decreasing and periodic sequences (A-level only), 1.6.11 Convergent and divergent sequences (A-level only), 1.6.12 Arithmetic sequences and series (A-level only), 1.6.13 Standard formulae for arithmetic series (A-level only), 1.6.14 Geometric sequences and series (A-level only), 1.6.15 Standard formulae for geometric series (A-level only), 1.6.16 Sum to infinity of a geometric series (A-level only), and 1.6.17 Sequences and series in modelling (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.