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## A Sequence

Math Teaser!

If the first five terms of a sequence are 0, 2, 8, 18, 32 what would be the value of the tenth term?

Start by looking at how each term in the sequence is related to the others. Since the terms grow bigger and bigger by an increasing margin with each increment, it makes sense that something with exponents might be at play. Look at each term’s index, or count number, and see if there is some exponent value that relates to each term. 0 could be squared and still result in 0, 1 could be squared – but how to get it to 2? Multiply by 2? Try this with a 2. 2 squared is 4, then multiplied by 2 is 8. Try once more with 3 – 3 squared is 9, multiply by 2 to get 18.

This looks right, so if this is the sequence relationship, we now need to figure out the value of the tenth term. Be careful, what we used for 0 was actually the first term (or n=1), so we need to account for that difference in the equation that describes the sequence. The equation is then:

$2 (n-1)^2 \text {, where n is the index value of the term}$

So, looking for the tenth term, or n=10, solve

$2 (10-1)^2$

to get 162.