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Summary

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1.7 Summary (EMCF5)

Arithmetic sequence

  • common difference \((d)\) between any two consecutive terms: \(d = T_{n} - T_{n-1}\)
  • general form: \(a + (a + d) + (a + 2d) + \cdots\)
  • general formula: \(T_{n} = a + (n - 1)d\)
  • graph of the sequence lies on a straight line

Quadratic sequence

  • common second difference between any two consecutive terms
  • general formula: \(T_{n} = an^{2} + bn + c\)
  • graph of the sequence lies on a parabola

Geometric sequence

  • constant ratio \((r)\) between any two consecutive terms: \(r = \frac{T_{n}}{T_{n-1}}\)
  • general form: \(a + ar + ar^{2} + \cdots\)
  • general formula: \(T_{n} = ar^{n-1}\)
  • graph of the sequence lies on an exponential curve

Sigma notation

\[\sum_{k = 1}^{n}{T_{k}}\]

Sigma notation is used to indicate the sum of the terms given by \(T_{k}\), starting from \(k =1\) and ending at \(k = n\).

Series

  • the sum of certain numbers of terms in a sequence
  • arithmetic series:
    • \(S_{n} = \frac{n}{2}[a + l]\)
    • \(S_{n} = \frac{n}{2}[2a + (n - 1)d]\)
  • geometric series:
    • \(S_{n} = \frac{a(1 - r^{n})}{1 - r}\) if \(r < 1\)
    • \(S_{n} = \frac{a(r^{n} - 1)}{r-1}\) if \(r > 1\)

Sum to infinity

A convergent geometric series, with \(- 1 < r < 1\), tends to a certain fixed number as the number of terms in the sum tends to infinity.

\[S_{\infty} = \sum_{n =1}^{\infty}{T_{n}} = \frac{a}{1 - r}\]