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 A Level Maths 1.4 Sequences and Series, covering 1.4.1 Binomial expansion for positive integer n, 1.4.2 Link to binomial probabilities, 1.4.3 Binomial expansion for rational n (A-level only), 1.4.4 Validity of the expansion (A-level only), 1.4.5 Sequences (A-level only), 1.4.6 Increasing, decreasing and periodic sequences (A-level only), 1.4.7 Sigma notation (A-level only), 1.4.8 Arithmetic sequences and series (A-level only), 1.4.9 Geometric sequences and series (A-level only), 1.4.10 Sum to infinity of a geometric series (A-level only), and 1.4.11 Modelling with sequences and series (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.