A physicist models the spectral density S(λ)S(\lambda)S(λ) of a laser beam for small wavelengths λ \lambda\,λ using the function
S(λ)=(14−12λ)−52∣λ∣<12S(\lambda) = \left( \frac{1}{4} - \frac{1}{2}\lambda \right)^{-\frac{5}{2}} \quad |\lambda| < \frac{1}{2}S(λ)=(41−21λ)−25∣λ∣<21
Determine the first 4 terms of the binomial expansion of S(λ)S(\lambda)S(λ) in ascending powers of λ\lambdaλ, simplifying each coefficient.
A secondary filter is applied such that the resulting density R(λ)R(\lambda)R(λ) satisfies
(14−12λ)nS(λ)=(14−12λ)−12\left( \frac{1}{4} - \frac{1}{2}\lambda \right)^{n} S(\lambda) = \left( \frac{1}{4} - \frac{1}{2}\lambda \right)^{-\frac{1}{2}}(41−21λ)nS(λ)=(41−21λ)−21
State the value of nnn.
Hence, or otherwise, find the first 3 terms of the binomial expansion of R(λ)R(\lambda)R(λ) in ascending powers of λ\lambdaλ, giving each term in its simplest form.
Practise Edexcel A Level Maths Binomial Expansion with exam-style questions for A Level Maths. 100 questions covering 4.1 Expanding (1 + x)^n, 4.2 Expanding (a + bx)^n, and 4.3 Using Partial Fractions, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.