In the calibration of a precision acoustic sensor, the signal response R(s)R(s)R(s) for a sensitivity parameter s s\,s is modeled by the function
R(s)=(2+ks4)5 R(s) = \left( 2 + \frac{ks}{4} \right)^5 R(s)=(2+4ks)5where k k\,k is a non-zero constant.
Find the first 4 terms, in ascending powers of sss, of the binomial expansion of R(s)R(s)R(s). Give each term in its simplest form.
An enhanced signal model is defined by
G(s)=(5+2s)(2+ks4)5 G(s) = (5 + 2s) \left( 2 + \frac{ks}{4} \right)^5 G(s)=(5+2s)(2+4ks)5In the series expansion of G(s)G(s)G(s), the coefficient of s s\,s is equal to the constant term. Find the value of kkk.
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.