In a materials science laboratory, the expansion factor of a synthetic polymer PPP relative to temperature change ttt is modeled by the function:
P(t)=(2+kt4)5 P(t) = \left(2 + \frac{kt}{4}\right)^5 P(t)=(2+4kt)5where kkk is a non-zero constant.
Find the first 4 terms, in ascending powers of ttt, of the binomial expansion of P(t)P(t)P(t). Give each term in its simplest form.
To account for sensor drift, a corrected expansion model R(t)R(t)R(t) is defined as:
R(t)=(5+3t)P(t) R(t) = (5 + 3t)P(t) R(t)=(5+3t)P(t)Given that in the expansion of R(t)R(t)R(t), the coefficient of the ttt term is equal to the constant term, determine 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.