Question

why is (1 + 2 + 3 + ..+ (n-2) ) not the same as (1 + 2 + 3 + ... + n) - 2? where n is the index that starts at 1?

Answer 1

I was trying to find a formula for a(n) that is define as a(1) = 1,a(n) = a(n-1) + n

which just turns out to be the sum of the naturals up to n

Answer 2

The first sum is telling you you are summing all numbers until the (n-2)th number. The second sum says you sum until the n-th number, then subtract 2.

An easier to see this is take n=7:

Answer 3

Sum 1:

1 + 2 + 3 + 4 + 5 (we stop at 5 since 5 = n - 2 = 7 - 2
=15

Sum 2:

(1 + 2+ 3 + 4 + 5 + 6 + 7) - 2 = 26

Answer 4

\[
(1 + 2 + 3 + ... + n) - 2 \\
= (1 + 2 + 3 + ... + (n-2) + (n-1) + n) - 2\\
= 1 + 2 + 3 + ... + (n-2) + 2n-3
\]

Answer 5

well yes, I realized that but what I don't get is, wouldn't the associative law of addiction enable us to do (1 + 2 + ... + n) - 2?

Answer 6

The reason why you have this is because of how sums work. You add consecutively in order. If you are familiar with summation notation, the first sum would be:
\[\sum_{i=1}^{n-2}i\]

The 2nd sum would be
\[\sum_{i=1}^{n}i - 2\]

Answer 7

maybe it's because n is an index?

Answer 8

sourwing, does what I wrote make sense?

Answer 9

It comes down to the peculiarity of the ellipse notation.

Answer 10

What happens is that unfortunately, the dots do not indicate that you have an equal number of terms. What it says "you can complete this sum up to the next term written".

In sum 1: you have 1 + 2 + 3 + (complete sum until (n-2)

In sum 2: you have 1 + 2 + 3 + (complete sum until n) and then - 2

Answer 11

The point is though that when doing these types of sums, the dots " ... " are a way of telling you how to do the summation. You also notice that we have (n-2) at the end of Sum 1. It is in parentheses on purpose so that you know you are summing until (n-2).

Although parenthesis have "no importance" in "regular" addition like
3 + (n - 2) is the same as 2 + n - 2

However, here the " ... " are a way of telling you how many terms you should be summing up. Why do we include them? Because often we have sums that go on for many terms (like 20 for example) and it's tedious to write every single term. Sometimes, we don't know how many terms we are summing (this is when we generally use \(n\) to represent this "any number". Also, sometime we are summing an infinite amount of terms.

Answer 12

@wio, yeah it does make sense

Answer 13

When you change the last term, you are changing the pattern, thus you are changing the number of terms.

Answer 14

3 + (n - 2) is the same as 3 + n - 2** sorry typos @_@

Answer 15

@kirbykirby thank you for the explanation. Those ellipse really mess things up because every one uses them without really defining what they actually mean

Answer 16

The ellipse generally are unambiguous.

Answer 17

they are.

Thanks guys ^.^

Answer 18

If you ever see \(+\dots +\) that means "this is a arithmetic sequence". anyone using them not to mean this is not using them correctly.

Answer 19

It just takes a bit of time to get used to. I was a bit confused too when I started seeing that.