<p>This was the last question in the section on my practice SAT test. Can someone please explain to me how to figure it out? Preferably an algebraic way?</p>
<p>The least integer of a set of consecutive integers is -25. If the sum of these is 26, how many integers are in this set?</p>
<p>Well, the integers are in consecutive order. That is the key. So, they will go up towards the positive numbers. So, you need to go to positive 25 until all the negatives cancel out with the positives. Then, once you get to 26 you’re sum will be 26. </p>
<p>25 + 1 + 25 + 1. I believe the answer is 52. (1 for the ‘0’ and the other 1 is for the ‘26’)</p>
<p>is it 52? </p>
<p>10char</p>
<p>Oh! Okay I get it now. I just forgot to add the 0.</p>
<p>Yes, the answer is 52.</p>
<p>Thanks!</p>
<p>Imagine a number line. The integer -25 is 25 units to the left of 0. The integer 25 is 25 units to the right of 0. This means that when you add them together you get 0. The same goes for -24 and 24; -23 and 23, etc. Here’s the beginning of the list of the consecutive integers in the set that the question is talking about:
-25, -24, -23, -22, -21, . . .
Obviously if you start off adding the numbers at the beginning, the sum starts as a negative number, which means that, in order to make the sum 26 (as the question says), you would have to cross over 0 and into the positive numbers. Since -25 and 25 make 0, -24 and 24 make 0, etc., the sum of all the numbers would be 0 if the set looks like this:
[-25, -24, -23, -22, -21, . . . , 21, 22, 23, 24, 25]; SUM = 0
For the sum to be 26, you would need to continue the set with one more number:
[-25, -24, -23, -22, -21, . . . , 21, 22, 23, 24, 25, 26]; SUM = 26</p>
<p>I don’t think this is a problem that has much to do with algebra. It’s conceptual.</p>
<p>EDIT: woops. I was late. And I didn’t even answer the question</p>