We think you are located in South Africa. Is this correct?

Summary

Do you need more Practice?

Siyavula Practice gives you access to unlimited questions with answers that help you learn. Practise anywhere, anytime, and on any device!

Sign up to practise now

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.

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