Link to Using the CASIO FX 82 ZA to solve Grade 12 Exam Questions
2022 TERM 1
| Week 1 | Week 2 | Week 3 | Week 4 | Week 5 | Week 6 | Week 7 | Week 8 | Week 9 | Week 10 |
|---|---|---|---|---|---|---|---|---|---|
| Number Patterns, Sequences and Series | Euclidean Geometry | Trigonometry | |||||||
Sequences and Series
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
- $a=T_1$
- $d=T_2-T_1=T_3-T_2$
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
- $T_1=a+b+c$…1st term
- $T_2-T_1=3a+b$….1st difference
- $(T_3-T_2)-(T_2-T_1)=2a$….2nd difference
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
- $a=T_1$
- $r=\frac{T_2}{T_1}=\frac{T_2}{T_1}$…constant ratio
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.
\[S_{\infty} = \sum_{n =1}^{\infty}{T_{n}} = \frac{a}{1 – r}\]