The signal noise N N\,N in a high-precision sensor is modeled by the function N(c)=(3+c4)9\displaystyle N(c) = \left(3 + \frac{c}{4}\right)^9N(c)=(3+4c)9, where c c\,c represents the concentration of a dopant in parts per million (ppm).
Find, in ascending powers of ccc, up to and including the term in c3c^3c3, the binomial expansion of N(c)N(c)N(c), fully simplifying each coefficient.
Use your expansion from part (a) to find an approximation for 3.02593.025^93.0259. Give your answer to 2 decimal places.
Without calculating the exact value of 3.02593.025^93.0259, state, with a reason, whether the answer to part (b) is an underestimate or an overestimate.
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.