1.7 Summary
Previous
1.6 Infinite series
|
Next
End of chapter exercises
|
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.
\[S_{\infty} = \sum_{n =1}^{\infty}{T_{n}} = \frac{a}{1 - r}\]
Previous
1.6 Infinite series
|
Table of Contents |
Next
End of chapter exercises
|