Notes
Sequences
| Arithmetic | Geometric | |
|---|---|---|
| Pattern | add $d$ | multiply by $r$ |
| Explicit | $a_n = a_1 + (n - 1)d$ | $a_n = a_1 r^{n - 1}$ |
| Recursive | $a_n = a_{n - 1} + d$ | $a_n = r \cdot a_{n - 1}$ |
Sigma notation
$\displaystyle \sum_{k=1}^{n}$ means plug in $k = 1, 2, \dots, n$ and add. The number of terms is top $-$ bottom $+ 1$.
Series formulas
- Arithmetic: $\displaystyle S_n = \frac{n(a_1 + a_n)}{2}$.
- Finite geometric: $\displaystyle S_n = \frac{a_1(1 - r^n)}{1 - r}$.
- Infinite geometric: if $|r| < 1$, $\displaystyle S = \frac{a_1}{1 - r}$; if $|r| \ge 1$, it diverges.
Common mistakes
- Using $n$ instead of $n - 1$ in the explicit formulas.
- Confusing the last term with the number of terms.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$-3 + 29(5) = 142$.
$\displaystyle r = \frac{1}{4}$, so $\displaystyle a_6 = 1024\left(\frac{1}{4}\right)^5 = \frac{1024}{1024} = 1$.
First term 2, last term 59: $\displaystyle \frac{20(2 + 59)}{2} = 610$.
$\displaystyle \frac{10}{1 - 0.6} = \frac{10}{0.4} = 25$.
Its ratio is $1.5$, and $|1.5| \ge 1$.
Geometric series: $\displaystyle \frac{40000(1 - 1.03^{10})}{1 - 1.03} \approx 458555$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A ball is dropped from 10 feet. After each bounce it rises to 80% of the height it fell from.
(a) Write a formula for the height $h_n$ of the $n$th rebound, and find the height of the 3rd rebound.
Show the answer and scoring guide
$h_n = 10(0.8)^n$. $h_3 = 10(0.512) = 5.12$ feet.
Check off the points you earned:
(b) Find the total of the heights of the first 5 rebounds.
Show the answer and scoring guide
$\displaystyle S_5 = \frac{8(1 - 0.8^5)}{1 - 0.8} = 40(1 - 0.32768) \approx 26.89$ feet.
Check off the points you earned:
(c) If the ball bounces forever, what total vertical distance does it travel?
Show the answer and scoring guide
It falls 10 feet, then each rebound is traveled up and down. Rebounds sum to $\displaystyle \frac{8}{1 - 0.8} = 40$. Total $= 10 + 2(40) = 90$ feet.
Check off the points you earned: