EIV - Formulas Strategy

[brandonb]

Could someone please help explain how the book came up with the following example?

Defining Rules for Sequences
Pg 39 - next to last paragraph

Therefore, the formula for this sequnce is n^2 + 6n + 11.

How did they compute this from the given sequence?
18
27
38
51

Thanks!

[StaceyKoprince]

hi, Brandon - in future, please post the entire text of the question. We don't always have access to our books when we're answering questions. Thanks!

I happen to have this book with me, though, so I can answer your question. (Also, this is on page 59, not page 39, for anyone else reading this.)

Standard form of a quadratic equation is y-sub-n = an^2 + bn + c, where a, b, and c are constants, n is the number of the term, and y is the value of that term. This standard formula applies whenever the difference between the difference between successive terms in a sequence is the same. That's confusing, I know. Basically that means this:
18
27
38
51

Diff bet 18&27 = 9
Diff bet 27&38 = 11
Diff bet 38&51 = 13

So the first-level difference is NOT the same (9, 11, and 13 are not the same number). If the first-level difference HAD been the same, we would've had an arithmetic sequence, for which the standard equation is kn + x. But it's not here, so check the second-level difference.

Diff bet 9&11 = 2
Diff bet 11&13 = 2

So the second-level difference IS the same. The standard equation, therefore is an^2 + bn + c (this is just something you'll have to memorize - though, FYI, these aren't that common on the test, so you'll have to decide whether you want to make the effort).

The original numbers in the sequence (18, 27, etc) are solutions for the sequence. 18 is the first term in the sequence, so y-sub-n = 18 when n = 1. y-sub-n = 27 when n = 2. And so on. I can substitute these into the standard equation, as is done in the 3rd to last paragraph on page 59. From there, I have three equations with three unknowns, so I can solve for the three unknowns, a, b, and c, and then plug them into the standard form of the equation. (Though it's unlikely you'd have to do this on the test b/c this would take anyone more than 2 minutes!)