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Algebraic Intuition

\begin{align*} \large 9^a&\large= 25 \\ \\ \large 3^a &\large = \ ? \end{align*}

What goes at the question mark in the equations above?

Hint: There's a much faster way than solving for $$a.$$ You are not a robot, and you don't need a calculator to solve this!

$\large \frac { 1 }{ { 2 }^{ 3 } } +\frac { 1 }{ { 2 }^{ 6 } } +\frac { 1 }{ { 2 }^{ 9 } } + \cdots = \ ?$

Hint: Don’t try this one on a calculator either - your fingers will wear out! This algebra problem can be solved with a geometric shortcut:

What total fraction of this image is red?

Which is greater, $$A$$ or $$B?$$

$A=\frac{99^{999}+1}{99^{1000}+1}$

$B=\frac{99^{1000}+1}{99^{1001}+1}$

$$\log_2 3$$ $$\times \log_3 4$$ $$\times \log_4 5$$ $$\times \log_5 6$$ $$\times \log_6 7$$ $$\times \log_7 8 = \, ?$$

It’s a race! Who is the winner? Choose wisely...

Which of these sequences will exceed $$1,000,000^{1,000,000}$$ first?

 Sequence A: $$1^2$$ $$(1^2)^3$$ $$((1^2)^3)^4$$ $$(((1^2)^3)^4)^5$$ ... Sequence B: $$2^1$$ $$(3^2)^1$$ $$((4^3)^2)^1$$ $$(((5^4)^3)^2)^1$$ ... Sequence C: $$2^1$$ $$3^{2^1}$$ $$4^{3^{2^1}}$$ $$5^{4^{3^{2^1}}}$$ ...

Note: Towers of exponents are evaluated from the top down, so $$3^{2^1} = 3^{\left(2^1\right)}.$$

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