# A Preview of More

An enormous diversity of problems can be solved by creatively applying algebraic techniques. This quiz showcases some of that diversity, pulling problems from five of the topics that we’ll cover in depth in this course:

• Factorials
• Rates & Ratios
• Sequences and Series
• Proving Algebraic Identities
• Avoid Proving that 1=2

# A Preview of More

Factorials

$\large \frac {{\color{red}6!} \times \color{blue}7!}{\color{purple}{10}!} = \, \color{green}?$

Note: The exclamation points are all factorials: $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1.$

# A Preview of More

Rates & Ratios

It takes 4 cooks 4 hours to bake 4 cakes. Assuming no change of rate, how many hours will it take 8 cooks to bake 8 cakes?

# A Preview of More

Sequences and Series

How many line-segments are there in the $$4^\text{th}$$ image of this growing tree sequence?

# A Preview of More

Proving Algebraic Identities

What is the sum of the first 13 positive odd numbers?

A big hint:

# A Preview of More

Help Us Avoid Proving That 1=2

Let $$a = b$$.

Then, $\begin{array} {c r c l } \color{red} {1} & ab & = & a^2 \\ \color{orange} {2} & ab - b^2 & = & a^2 -b^2 \\ \color{gold} {3} & b(a-b) & = & (a+b)(a-b) \\ \color{green} {4} & b & = & a+b \\ \color{blue} {5} & b & = & b+b \\ \color{indigo} {6} & b & = & 2b \\ \color{purple} {7} & 1 & = & 2. \end{array}$

Which line is the first that contains an error in this proof?

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