Notes
Sequences and series
- Arithmetic: $a_n = a_1 + (n - 1)d$ and $\displaystyle S_n = \frac{n(a_1 + a_n)}{2}$.
- Geometric: $a_n = a_1 r^{n-1}$ and $\displaystyle S_n = \frac{a_1(1 - r^n)}{1 - r}$. If $|r| < 1$, the infinite sum is $\displaystyle \frac{a_1}{1 - r}$.
- Sigma notation $\displaystyle \sum_{k=1}^{n} a_k$ adds the terms from $k = 1$ to $k = n$.
Composition and inverses
- $(f \circ g)(x) = f(g(x))$: apply $g$ first. Order matters.
- To find $f^{-1}$: swap $x$ and $y$, then solve for $y$. If $f(a) = b$ then $f^{-1}(b) = a$.
Choosing a model
- Constant first differences: linear. Constant second differences: quadratic. Constant ratios: exponential.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$a_{20} = 5 + 19(4) = 81$ and $\displaystyle S_{20} = \frac{20(5 + 81)}{2} = 860$.
$a_5 = 5 \cdot 3^4 = 405$.
$\displaystyle r = \frac{1}{2}$, so $\displaystyle S = \frac{12}{1 - \frac{1}{2}} = 24$.
$g(4) = 8$, then $f(8) = 11$.
Swap: $\displaystyle x = \frac{y - 1}{4}$, so $y = 4x + 1$.
Each output is 2 times the previous one.
Free response
Write your answer first, then open the scoring guide.
Free-response question
Maya saves $20 in week 1 and increases her deposit by $5 each week. Jon saves $1 in week 1 and doubles his deposit each week.
(a) How much does Maya deposit in week 10?
Show the answer and scoring guide
Maya's deposits are arithmetic with $d = 5$: $a_{10} = 20 + 9(5) = 65$. She deposits $65.
Check off the points you earned:
(b) What is the total amount Maya has saved after 10 weeks?
Show the answer and scoring guide
$\displaystyle S_{10} = \frac{10(20 + 65)}{2} = 425$. She has saved $425.
Check off the points you earned:
(c) What is Jon's total savings after 10 weeks, and who has saved more?
Show the answer and scoring guide
Jon's deposits are geometric with $r = 2$: $\displaystyle S_{10} = \frac{1(1 - 2^{10})}{1 - 2} = 1023$. He has saved $1,023, which is more than Maya's $425.
Check off the points you earned: