Algebraic Thinking
8.1 Variables
Thinking with Variables
Algebra begins when we stop thinking only about specific numbers and start describing general patterns.
Definition: Constant
A constant is a fixed value.
Definition: Variable
A variable is a symbol that can represent one or more values.
Example: Constants vs variables
Identify constants and variables in each formula.
- $A=lw$
- $P=2l+2w$
- $C=2\pi r$
- $F=\frac95C+32$
- $A=Pe^{rt}$
Show Solution
- Variable symbols: $A,l,w$. Constants: implied 1.
- Variables: $P,l,w$. Constants: 2.
- Variables: $C,r$. Constants: 2 and $\pi$.
- Variables: $F,C$. Constants: $\frac95$ and 32.
- Variables: $A,P,r,t$. Constants: $e$.
The same symbol can be variable in one context and fixed in another; context decides role.
Example: Translate words to algebra
A car rental costs a flat $50 dollars plus 0.20 dollars per mile m. Write the cost expression.
Show Solution
Flat fee plus per-mile fee gives
$$C(m)=50+0.20m.$$
That expression can now be used for any mileage input.
MyOpenMath: MOM: Variables and translation
Arithmetic Sequences as Functions
An arithmetic sequence has constant difference d, so terms grow linearly.
Fact: Arithmetic n-th Term Formula
If first term is $a_1$ and common difference is $d$, then
$$a_n=a_1+d(n-1).$$
Example: Build sequence from two terms
An arithmetic sequence has $a_2=11$ and $a_5=23$. Find d, formula, and first terms.
Show Solution
Use term formula twice:
$$a_2=a_1+d=11$$ $$a_5=a_1+4d=23.$$
Subtract equations: $3d=12$, so $d=4$. Then $a_1=11-4=7$.
So
$$a_n=7+4(n-1)=4n+3.$$
First terms are 7, 11, 15, 19, 23, ...
Example: Another arithmetic reconstruction
Given $a_{10}=100$ and $a_{15}=145$, find formula and $a_{500}$.
Show Solution
Set up:
$$a_1+9d=100, \quad a_1+14d=145.$$
Subtract: $5d=45\Rightarrow d=9$. Then $a_1=100-81=19$.
Formula:
$$a_n=19+9(n-1)=9n+10.$$
Then
$$a_{500}=9(500)+10=4510.$$
MyOpenMath: MOM: Arithmetic sequences
Geometric Sequences as Functions
A geometric sequence has constant ratio r, so terms grow exponentially.
Fact: Geometric n-th Term Formula
If first term is $a_1$ and ratio is $r$, then
$$a_n=a_1r^{n-1}.$$
Example: Geometric sequence from two terms
Given $a_2=3$ and $a_5=81$, find formula and $a_{10}$.
Show Solution
From formula:
$$a_2=a_1r=3, \quad a_5=a_1r^4=81.$$
Divide equations: $r^3=\frac{81}{3}=27$, so $r=3$. Then $a_1=\frac{3}{3}=1$.
So
$$a_n=3^{n-1}.$$
Then
$$a_{10}=3^9=19683.$$
Example: Growth model
A bacteria culture doubles every 3 hours and starts at 500. Write formula and find amount after 24 hours.
Show Solution
In 24 hours there are $\frac{24}{3}=8$ doubling periods.
$$B(t)=500\cdot 2^{t/3}.$$
So
$$B(24)=500\cdot 2^8=500\cdot 256=128000.$$