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Math and Logic

What Does Average Look Like?

Representing an average visually can help us shorten or even completely avoid calculations. For example, when we're finding the average of two numbers, we can also represent it visually using a number line.

If two numbers \(a\) and \(b\) have an average of \(c,\) then that value will lie exactly halfway between them on a number line.

This process can be repeated, and it can even be used if we know different bits of information.

For example, suppose we know that the average of 8 and some number \(j\) is 24. Representing this visually gives us the following image:

Since we know that the average, 24, must be exactly in the middle of 8 and \(j\), the distance from either of them is equal. And since we can see that the distance between 8 and 24 is 16, \(j\) must also be 16 units away from 24, but on the other side. Thus, \(j\) must be \(24+16= 40\).

In the problem below, there's enough information to solve as a (fairly involved) system of equations—just write down in algebraic form everything you know about the numbers listed, and you'll eventually be able to find all the values. However, thinking about averages visually can get you to the same answer in a very different (and more efficient) way.

Today's Challenge

In the sequence of integers below, each number (starting with \(\small b\)) is the average of the two numbers before it. What is the value of \(a?\)

\[{45, a, b, c, d, 100}\]

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