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1.6 Sequences and Series

1.6 Sequences and Series

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Question 128

A theoretical physicist is studying the relativistic correction factor, C(v)C(v)C(v), for a particle's effective mass at different velocities vvv. The factor is modeled by the function C(v)=116+vC(v) = \frac{1}{\sqrt{16 + v}}C(v)=16+v​1​.

a.

Find the first three terms, in ascending powers of vvv, of the binomial expansion of C(v)C(v)C(v).

[4]
b.

Hence, find the first three terms of the binomial expansion of C(−v2)=116−v2C(-v^2) = \frac{1}{\sqrt{16 - v^2}}C(−v2)=16−v2​1​.

[2]
c.

Using your answer to part (b), find an approximation for ∫02116−v2 dv\int_{0}^{2} \frac{1}{\sqrt{16 - v^2}} \, dv∫02​16−v2​1​dv, giving your answer to seven decimal places.

[3]
d.

(i) A research assistant decides to use this method to find a more accurate value for the integral by increasing the number of terms of the binomial expansion used. Explain clearly whether this new approximation will be an overestimate, an underestimate, or if it is impossible to tell.

(ii) The assistant goes on to use the expansion from part (b) to find an approximation for ∫05116−v2 dv\int_{0}^{5} \frac{1}{\sqrt{16 - v^2}} \, dv∫05​16−v2​1​dv. Explain why this calculation is invalid.

[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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