# Question #bf489

Mar 1, 2017

$y = 2 x + 2$

#### Explanation:

Observe that:
Difference in each step for advertising is +2
Difference in each step for revenue is +4

Let the advertising signature be $i$
Let the ith advertising signature be ${a}_{i}$

$\textcolor{b l u e}{\text{Consider the number structure}}$

${a}_{0} \to 2$
${a}_{2} \to 6$
${a}_{4} \to 10$
${a}_{6} \to 14$
${a}_{8} \to 18$
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We need to relate $i$ to revenue. ${a}_{0} \to \left(0 - 2\right) + 4 = 2$

$\textcolor{red}{{a}_{0} \to - 2 + 4 = 2}$
$\textcolor{red}{{a}_{2} \to - 2 + 4 + 4 = 6}$
$\textcolor{red}{{a}_{4} \to - 2 + 4 + 4 + 4 = 10}$
$\textcolor{red}{{a}_{6} \to - 2 + 4 + 4 + 4 + 4 = 14}$
$\textcolor{red}{{a}_{8} \to - 2 + 4 + 4 + 4 + 4 + 4 = 18}$
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Lets try something:

$\textcolor{red}{{a}_{0} \to - 2 + 4 = 2}$
$\textcolor{g r e e n}{{a}_{1} \to - 2 + 4 + 2 = 4}$
$\textcolor{red}{{a}_{2} \to - 2 + 4 + 2 + 2 = 6}$
$\textcolor{g r e e n}{{a}_{2} \to - 2 + 4 + 2 + 2 + 2 = 8}$
$\textcolor{red}{{a}_{4} \to - 2 + 4 + 2 + 2 + 2 + 2 = 10}$
$\textcolor{g r e e n}{{a}_{5} \to - 2 + 4 + 2 + 2 + 2 + 2 + 2 = 12}$
$\textcolor{red}{{a}_{6} \to - 2 + 4 + 2 + 2 + 2 + 2 + 2 + 2 = 14}$
$\textcolor{g r e e n}{{a}_{7} \to - 2 + 4 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 16}$
$\textcolor{red}{{a}_{8} \to - 2 + 4 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 18}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This is the same as:
${a}_{0} \to 2 = 2$
${a}_{1} \to 2 + 2 = 4$
${a}_{2} \to 2 + 2 + 2 = 6$
${a}_{3} \to 2 + 2 + 2 + 2 = 8$
${a}_{4} \to 2 + 2 + 2 + 2 + 2 = 10$
${a}_{5} \to 2 + 2 + 2 + 2 + 2 + 2 = 12$
${a}_{6} \to 2 + 2 + 2 + 2 + 2 + 2 + 2 = 14$
${a}_{7} \to 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 16$
${a}_{8} \to 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 18$

so any ${a}_{i} = 2 \left(i + 1\right) = 2 i + 2$

As this is linear we use the standardised format $y = m x + c$

Set $i = x$ giving: $y = 2 x + 2$

I will let you draw the graph