Math CentralQuandaries & Queries


Question from Dawn, a student:

Find an expression for the nth term:

x= 1 2 3 4...n
y=11 8 5 2...?

I think it's a linear relation but don't know the equation that goes with it.

Dawn, you are right. That's a linear relation.

Look for the patterns: each x increases by one for successive terms and each y decreases by 3 similarly.

So you can get the value of y for any corresponding x by turning the pattern into an equation like this:

When x = 1, then y = 11 (that's the starting point).

But if x is larger, say 15, then it has gone up by 15-1 = 14 steps. So y has gone down 14 times. Each time it goes down, it drops by 3, so it goes down by 3 times 14, which is 42. Since it started at 11, it goes to 11 - 42 = -31.

That's the same as saying "take x and subtract 1. Then multiply that by 3. Then subtract that from the starting value of y (that's 11). That's the new value of y."

Turn that into an equation:

"Take x and subtract 1" = x - 1.
"Then multiply that by 3" = 3(x - 1).
"Then subtract that from 11" = 11 - 3(x - 1).
"That's the new value of y" = y.

y = 11 - 3(x - 1).

This simplifies: y = 14 - 3x.

Check it with x = 15. What value of y were you expecting?

Hope this helps,
Stephen La Rocque.

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