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1.6 Sequences and Series

1.6 Sequences and Series

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Question 212

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∣<12 S(\lambda) = \left( \frac{1}{4} - \frac{1}{2}\lambda \right)^{-\frac{5}{2}} \quad |\lambda| < \frac{1}{2} S(λ)=(41​−21​λ)−25​∣λ∣<21​
a.

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.

[4]
b.

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​−21​λ)nS(λ)=(41​−21​λ)−21​

State the value of nnn.

[1]
c.

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.

[3]
Markscheme

1.6 Sequences and Series Questions

  1. A Level
  2. /Maths
  3. /1.6 Sequences and Series

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.

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