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

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.