Sum with three notations around it.

Question

Have seen the below notation (How to calculate number of triangles and points after dividing a triangle n times?) and need to break it down into plain english so to speak. This just so I can catch up.

$$ P(n) = 3 + \sum_{k=1}^{n} 3^n$$

The capital sigma is for sum but then there is this $n$ above and $k=1$ below. If done with it what is needed it is propably multiplied by 3 to the power of $n$. Think this would be added to the 3 and that together would be $P(n)$

What P stands for in $P(n)$ unknonw to me. What the parenthesis are doing, don't know either.

Answer 1

This is a typo. It should probably be $P(n)=3+\sum_{k=1}^n 3^k$. $P$ is just some function, a way of defining various numbers, one for each number $n$. The parentheses just indicate that $n$ is what we're associating to $P$. So for instance the number $P(2)$ is $3+\sum_{k=1}^2 3^k=3+3+3^2=15$. But this explanation might not be clear: if you don't know what the parentheses are doing then you must not be familiar with the concept of a function, which is a bit tricky.

Answer 2

$P$ is a function, $n$ is the variable. I suppose the 3 is quite self-explanatory. $\sum_{k=1}^{n}{3^n}$ just means the sum of all values of $3^n$ as $k$ goes from 1 to $n$. So for example for $n=3$, the sum would be $3^3+3^3+3^3=81$, since there are three distinct values for $k$ as it goes from 1 to 3 (namely 1, 2, 3). Therefore$P(3)=3+81=84$.

Answer 3

The notation $$\sum_{k=1}^n 3^k$$ is a sort of a shorthand for $$3^1 + 3^2 + \cdots + 3^n$$ except that instead of requiring that you figure out what the $\cdots$ part should mean, it makes this explicit: You write the “$3^k$” part $n$ times, once with $k$ replaced by each integer from $1$ to $n$ inclusive, and then you add up the resulting $n$ expressions. For this example it's overkill, but for more complicated examples it is very useful because it removes the ambiguity inherent in the “$\cdots$”.

The $P(n)$ thing means that from now on, the notation $P(n)$ will be an abbreviation for $\displaystyle{3 + \sum_{k=1}^n 3^k}$, and so if the author writes something like “$P(\color{maroon}{7})$” you should understand this as meaning the same as $\displaystyle{3 + \sum_{k=1}^\color{maroon}{7} 3^k}$, which in turn is an abbreviation for $3 + 3^1 + 3^2 + 3^3 + 3^4 + 3^5 + 3^6 + 3^{\color{maroon}{7}}$, which happens to add up to 3732.

Having made this abbreviation mathematicians will then consider the entire process of starting with a number, such as $7$, and ending with another number, in this case 3732, and they will consider the process itself as a sort of mathematical object called a “function”. They will then refer to the function by the name “$P$”, and talk about the properties that the function $P$ has or doesn't have.

The original author wrote “$3^n$” rather than “$3^k$”. If this weren't an error, the expression would mean something slightly different, but since it's a mistake I'm not sure it would be worth your time to think about it.

I think that hits all your questions; if I missed anything please leave a comment.