12/11/09

Test 7 Section 7 - #18 (page 789)

When they say they're looking for the "sum of consecutive integers from -22 to x, inclusive, is 72" they mean this:

-22 + -21 + -20 + ... + x = 72

That's the way your math teacher would write it, anyway. :)

What you need to recognize here is that for a while, you're going to be getting nothing but negative numbers for your sum.  -22 + -21 + -20 = -63, for example.  And it's only going to get worse when you add in -19.  You're going to have to go a long way before your numbers start to go positive again.  Think for a minute about when these numbers will first start getting positive.  Or, even better: ask yourself when your sum is going to be zero.

-22 + -21 + -20 + ... + 20 + 21 +22 = 0

You'll have zero when you add up every consecutive integer from -22 to 22; the positive and negative numbers will cancel each other out.  From there, we're cruising.  Watch the bolded bits.

-22 + -21 + -20 + ... + 20 + 21 +22 + 23 = 23
-22 + -21 + -20 + ... + 20 + 21 +22 + 23 + 24 = 47
-22 + -21 + -20 + ... + 20 + 21 +22 + 23 + 24 + 25 = 72

The last number we had to add was 25, so that's our x.  (B) is our answer.

Of course, the long way is just to get crackin' with the calculator (if your fingers are fast it won't take that long) and start adding things up.  When you get to 72, you're done.  No shame in doing it that way, especially if it's the first way you see, and you start on it right away. 

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