1.7 Summary
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1.7 Summary (EMCF5)
Arithmetic sequence
 common difference \((d)\) between any two consecutive terms: \(d = T_{n}  T_{n1}\)
 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: \(r = \frac{T_{n}}{T_{n1}}\)
 general form: \(a + ar + ar^{2} + \cdots\)
 general formula: \(T_{n} = ar^{n1}\)
 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 \(r < 1\)
 \(S_{n} = \frac{a(r^{n}  1)}{r1}\) 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.
\[S_{\infty} = \sum_{n =1}^{\infty}{T_{n}} = \frac{a}{1  r}\]
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