4.2.10

Modelling with Sequences & Series

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Modelling with Sequences and Series

We can use sequences and series to model numerical problems that occur in real life.

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Fixed amount

  • Numbers that increase by fixed amounts can be modelled by arithmetic sequences.
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Example

  • On the Sunday after a minor surgery, a woman is able to walk a half-mile. Each Sunday, she walks an additional quarter-mile.
  • After 8 weeks, what will be the total number of miles she has walked?
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Arithmetic series

  • The number of miles the woman walks increases by a fixed amount each week.
  • This means we can model this as an arithmetic series with a1=0.5a_1 = 0.5 and d=0.25d = 0.25.
  • Using the equation for the sum of an arithmetic series, we have:
    • S8=12n(2a1+(n1)d)=82×(2×0.5+7×0.25)=11S_8 = \frac{1}{2}n(2a_1 + (n-1)d) = \frac{8}{2}\times(2\times0.5 + 7\times0.25) = 11
  • So the woman has walked a total of 11 miles after her surgery.
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Fixed percentage

  • Numbers that increase by a fixed percentage or fraction can be modelled by geometric sequences.
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Example

  • A pendulum travels a distance of 48 cm on its first swing. On each successive swing, it travels 34\frac{3}{4} of the distance of the previous swing.
  • What is the total distance travelled by the pendulum when it stops swinging?
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Geometric series

  • The distance of the swing decreases by a fixed fraction.
  • That means we can model total distance of the pendulum as a geometric series with a=48a = 48 and r=34r=\frac{3}{4}.
  • Since r<1|r|<1, we know that the infinite series is convergent and that the pendulum's distance will diminish until it eventually stops.
  • Finding the sum of the infinite series will give the total distance travelled by the pendulum.
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Sum of infinite series

  • Using the equation for the sum of an infinite geometric series, we have:
    • S=a1r=48134=4814=192S_{\infty}=\frac{\small a}{\small 1-r}=\frac{\small 48}{\small 1-\frac{3}{4}} = \frac{\small 48}{\small \frac{1}{4}}=192
  • So the pendulum travels a total distance of 192 cm before it stops.

Jump to other topics

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

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