My previous algebra course did not go over summation at all, and now that I'm in my new course, Discrete Math for Information Technology, we have been introduced to summation. I understand summation notation, but I have no idea how to solve equations that contain them. I have looked around on math.stackexchange.com and it seems everyone has specific equations they want solved. What I'd like is something more along the lines of a crash course in solving equations that contain summations. I'm not sure if this question is specific enough, so I'll also include the problem I'm working on.
"A lumberjack has $4n + 110$ logs in a pile consisting of $n$ layers. Each layer has two more logs than the layer directly above it. If the top layer has six logs, how many layers are there?"
I've worked out that each layer has $6+2(i-1)$ logs, where $i$ represents the current layer. I've put this into summation notation to represent all possible layers and set it equal to $4n +110$, which gives me:
$\displaystyle\sum\limits_{i=1}^n [6+2(i-1)]=4n+110$
I am now at a stage where all I have to do is solve for $n$, but I have no idea how to even start since I've not been taught how to solve equations that contain summations.
Best Answer
In your case, here are some results for you to either derive or prove through induction:
$$\sum_{i=1}^n 1 = n$$
$$\sum_{i=1}^n i = \frac{1}{2}n(n+1)$$
$$\sum_{i=1}^n i^2 = \frac{1}{6}n(n+1)(2 n+1)$$
$$\sum_{i=1}^n i^3 = \frac{1}{4}n^2(n+1)^2$$
Note that the summation is linear, so that for a term as you have above, you can break it up into
$$\sum_{i=1}^n 6 + \sum_{i=1}^n 2 i - \sum_{i=1}^n 2 = 6 \sum_{i=1}^n 1 + 2 \sum_{i=1}^n i - 2 \sum_{i=1}^n 1$$