I have difficulty counting Take this example:
$$84 + 87 + 90 + 93 .. + 180 + 183$$
If we want to use Gauss' way of finding this sum, we have to find the number of elements in this. What I do is just take the difference of the highest and lowest sum (99) and divide by 3, but this yields 33 and that can't be correct because it needs to be an even number in order to make sense. Why does it count 34 elements instead of the 33 my (faulty) arithmetic gives me?
 A: It's one of those typical "off by one" errors. The best way to see how to count the elements in such a sequence is to start with few elements: 


*

*If you have $84+87$ you have $(87-84)/3=1$ and there are $2$ elements in the sum  

*If you have $84+87+90$ you have $(90-84)/3=2$ and there are $3$ elements in the sum

*If you have $84+87+90+93$ you have $(93-84)/3=3$ and there are $4$ elements in the sum


I guess by now you see the pattern ...


*

*If you have $84+87+90+\cdots183$ you have $(183-84)/3=33$ and there are $34$ elements in the sum

A: Try finding $3+6+\cdots+183$, and then $3+6+\cdots+81$, and subtracting.
A: There are 34 terms starting with $84+0 \times 3=84$ and ending with $84+33\times 3=183$
A: We want
\begin{align}
\color{red}{S = 84 + 87 + 90 + 93 + \cdots + 180 + 183}
\end{align}
We also have
\begin{align}
\color{blue}{S = 183 + 180 + 177 + 174 + \cdots + 87 + 84}
\end{align}
This gives us
\begin{align}
\color{green}{2S} & \color{green}{= (84+183) + (87+180) + (90+177) + \cdots + (180+87) + (183+84)}\\
& \color{green}{= (267) + (267) + (267) + \cdots + (267) + (267)}\\
\end{align}
Let us count how many such $267$'s are there: We have $183 = 84 + 3k$. Note that $k+1$ gives the number of terms in the summation. We get $k = \dfrac{183-84}3 = 33$. Hence, there are $34$ terms. We hence get that
$$\color{green}{2S = 34 \times 267 \implies S = 17 \times 267 = 4539}$$
A: When you count $1$ through $10$ you have $10$ elements, but $10-1=9$. The extra one comes because you are counting both ends, while taking the difference counts the gaps between them. You have to add $1$ to account for that.
A: Every number in your series is divisible by 3, so we can write it as:
$$84+87+90+93..+180+183 = 3(28+29+...+60+61)$$
Know using Gauss formula we can get:
$$3(28+29+...+60+61) = 3(\frac{61*62}{2} - \frac{27*28}{2})$$
We can simplify this to:
$$3(\frac{34*62 + 27*62 - 27*28}{2}) = $$
$$3(\frac{34*62 + 27(62-28)}{2}) = $$
$$3(\frac{34*62 + 27*34}{2}) = $$
$$3(\frac{34(62+27)}{2}) = $$
$$3(\frac{34*89)}{2}) = $$
$$\frac{102*89}{2} = 4539$$
