"A Neat Trick!" solution

The trick to adding the integers 1 to 100 quickly is to do it twice. 
No, seriously. 
​But be creative in how you order the numbers, and write the second group in reverse order:
​
​\( (1+2+ \ldots +99+100)+(100+99+ \ldots +2+1) \)

Now, how does that help us?  Well, it doesn't….yet.  But if we arrange them a little differently into columns, and add each column first, we have 
\[ \array{ &1 & +& 2&+& \ldots &+ & 99 & + & 100&  \\  +&100&+&99&+& \ldots &+& 2 &+&1&\\ \hline \\ =&101&+&101&+& \ldots &+&101&+&101& } \]
​Do you see what we did?  We just rewrote the numbers so that all of the columns would be equal, and then summed each column.  Since we know there are \(100 \) columns, we can now use multiplication (much faster than addition) to see that the above is equal to \(100 \times 101=10,100 \).

But we're not done yet.  In order to make our columns equal, we had to add our sequence twice.  So now, to get our final answer, we need to divide \(10,100\) by \(2\) to get \(5050\). 

[And just remember, a 6-year-old figured this out!  Of course, that 6-year-old was none other than Carl Friedrich Gauss, one of the greatest mathematicians of all time.  And we really don't know that he was 6, we just know he was in primary school.  But still pretty impressive, wouldn't you say?]

Now, if you look closely you'll see that there's nothing special about the number 100 in this trick.  Any number will work.  And you don't even have to remember the steps we took.  Let's just make a formula, and then you'll have everything you need.

​General Formula
Suppose \(N \) is a positive integer, and we want to add the numbers \(1 \) through \(N \), written as \(1+2+ \ldots +(N-1)+N \)

As before, let's add them twice and line them up in columns:
​\[ \array{ &1 & +& 2&+& \ldots &+ & (N-1) & + & N&  \\  +&N&+&(N-1)&+& \ldots &+& 2 &+&1&\\ \hline \\ =&(N+1)&+&(N+1)&+& \ldots &+&(N+1)&+&(N+1)& } \]
This is equal to \((N+1) \times N \), and since we added everything twice to do our trick, we need to divide by 2.  And this gives our formula:
\[ 1+2+ \ldots +(N-1)+N= \frac{N \times (N+1)}{2} \].

​Go ahead and try it out!!  It works every time.  ​

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And here is some homework for you to try.
First, try this one:
1.  Calculate \(1+2+ \ldots +37+38 \).
Now let's see if you can adapt your thinking to these:
2.  Calculate \(2+4+ \ldots +98+100 \).
3.  Calculate \(20+21+22+ \ldots +79+80 \).
4.  Calculate \( 37+40+43+46+ \ldots +94+97+100\).

If you have trouble adapting the formula for those last three, you can always add them twice and line up the columns, right?

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