Consecutive Integers, guide 3 (word problems)

[JannaW716]

Hi everybody,
As I am working through the third strategy guide (word problems) I got confused about rules for consecutive integers.
How do page 109 and page 136 go together ? Don’t they say opposite things ?
Because on page 109 it says that the sum of an evenly spaced set is: Sum= average * number of terms
Where as on page 136 it says that for an even number ob consecutive integers the sum is never a multiple of the number of terms.

This is also part of problem 2 on page 139. I personally averaged the first and last term (=103,5) and multiplied it with the number of terms (100) to get the total sum. In that case the sum would be divisible by 100 though, while the solution says its not.

I would be very grateful is somebody could help with that!

Thank you

[Sage Pearce-Higgins]

Let's see if I can help clear this up. As you wrote, for a set of evenly spaced numbers: Sum= average * number of terms. But where does this come from? Take an example: {1, 2, 3, 4, 5}. Go on and add up these numbers the long way (1+2+3+...) and you'll get 15. Then notice that this is the same as the middle number (3) multiplied by the number of terms (5). The special thing about evenly spaced sets is that the middle number (that's the median) is the same as the mean.

Now take another example; {2, 3, 4, 5}. What's the sum of this set? Again, try it the long way, and then notice that the median here isn't actually one of the numbers. As there's an even number of terms the median is 3.5. There are 4 terms and the sum is 14, and 14 is not a multiple of 4. For sets of consecutive integers with an even number of terms, the median will always be a decimal. So the sum will be the number of terms multiplied by a decimal.

This is also part of problem 2 on page 139. I personally averaged the first and last term (=103,5) and multiplied it with the number of terms (100) to get the total sum. In that case the sum would be divisible by 100 though, while the solution says its not.

Your reasoning is good here. The sum is 103.5 * 100 = 10350. However, this number is not divisible by 100. Perhaps you're misunderstanding the term "divisible by 100". This means "gives an integer answer when divided by 100". Take a simpler example: 'Is 7 divisible by 2?' The answer is "no". Of course you can divide 7 by 2, but you get 3.5, not an integer answer. So we'd say that 7 is not divisible by 2.