Geometric Numbers

Notice that for every figure, the pink colored dots represent the newly added dots, while the black color dots represent the dots from the previous figure.

We notice that the added dots for the first $4$ figures are $1,4,7,10$

This implies that the number of dots added to each figure follow an arithmetic progression where $a=1,\;d=3$

Therefore, the number of dots added to the $37^{\text{th}}$ figure $=1+(37-1)(3) = 1+36(3) = 1+108=\boxed{109}$

Relevant wiki: General Term Pattern Recognition

The pentagonal number general formula is: $\large p_{n} = \frac{3n^{2} - n}{2}$

Therefore, the number that has to be added to get a next one is:

$\large{ p_{n+1} - p_{n} \\ = \quad \frac{3(n+1)^{2} - (n + 1)}{2} - \frac{3n^{2} - n}{2} \\ = \quad \frac{3(n^{2} + 2n + 1) - (n + 1) - (3n^{2} - n)}{2} \\ = \quad \frac{3n^{2} + 6n + 3 - n - 1 - 3n^{2} + n}{2} \\ = \quad \frac{6n + 2}{2} \\ = \quad 3n + 1 }$

which $n$ is the current pentagonal number, in this case, it's $36$ ; $n + 1$ is the next pentagonal number we want to get, in this case is 37.

Apply the formula, $n = 36$ , therefore, the number of points we have to add to get the 37th pentagonal number is $3 \times 36 + 1 = \fbox{109}$

3n+1 ... so to get the 37th pentagonal number, you have to add 3(36) + 1 = 109 p0ints.

Observe that the ${\text{n}}^{\text{th}}$ pentagonal number is of the form $n^2 + \displaystyle \sum_{i = 1}^{n-1} i$ .

So, the ${36}^{\text{th}}$ pentagonal number is ${36}^2 + \displaystyle \sum_{i = 1}^{35} i$ and the ${37}^{\text{th}}$ one is $37^2 + \displaystyle \sum_{i = 1}^{36} i$

So, the number of points that have to be added is:-
$\begin{aligned} {37}^2 + \displaystyle \sum_{i = 1}^{36} i - \left(n^2 + \displaystyle \sum_{i = 1}^{n-1} i\right) & = {37}^2 - {36}^2 + \dfrac{36 × 37}{2} - \dfrac{35 × 36}{2}\\ \\ & = \left(36+37\right)\left(37-36\right) + 18\left(37 - 35\right)\\ \\ & = 73 + 18×2\\ & = 73 + 36\\ & = \color{#3D99F6}{\boxed{109}} \end{aligned}$

The 1st pentagon has a side of 1 (it's a degenerate pentagon of size 1). The 2nd pentagon has a side of 2, the 3rd a side of 3, etc. The number of points added follows a pattern for an n-sided pentagon from an (n-1)-sided pentagon of n+n+(n-2).

The 37th pentagon will have a side of 37, and the number of points added will need to be 37+37+(37-2) = 109.

Here each edge has n dots.Now for pink dots there are 3 edges from 3rd. By adding all those in 3 edges we get 3n.But 2 dots we count twice so 3n-2.

I think another solution is n+2(n-1) Here it is 37+2(37-1)=109

Since this pattern is pentagonal, you always add three sides on the top of the previous one to continue the pattern. Each side has n points, and these three sides share two vertices. When counting points on each side, you will repeat twice of these two vertices. Thus, the difference between n-1 th and n th = 3n - 2. when n=37, difference = 3 x 37 - 2=109.

math is like the lunch table (differences matter), so let's make a chart: let's explain the columns: n counts the pentagons, where n=1 is our first pentagon and n=36 is the one we're asked about; f(n) counts the dots in pentagon #n; f'(n) counts how many dots we added from the last pentagon, and f''(n) counts how many dots we added to f'(n)

(goes on forever)

anyways, notice that the right column, f''(n) is constant for the first two. Let's assume that, if it was gonna change, it would've (theres 3 tries so being lazy is chill), which means f''(n)=3 (that's our conjecture, for now). If you've taken calculus, yk where this is going: you can integrate f''(n) to get f'(n)* and, if you do it right, you get f'(n)=3n+C, and we know f'(2)=7=3(2)+C, which means C is 1. So, we can write f'(n)=3n+1**which gives us the # of dots that are gonna get added as a function of n.

Plugging in n=36 (bc we wanna know what the jump is from 36 to 37), we get 3(36)+1=109, and that works! if you wanna challenge yourselves, find another way to interpret any of these columns (as a matrix or wtv idk), and take a look inside the centers of the pentagons.

*more specifically, you switch from n to x, where x varies over all real #s, not just whole ones, & integrate that. it's the same tho.

Each of the n-th figure has n sized boundary sides. There are three slanted sides which are added in a new figure. In the newly added sides, 2 end points coincide for the middle side.

So, the number of additional points in the n-th figure is - (3*n)-2.

The $n+1^{th}$ number in the sequence is equal to the $n^{th}$ number plus $3n+1$ .

Here we have $n = 36$ so calculating $3(36)+1$ gives us 109.

You can use simple equation $4 + 3(n-1) = x, n \geq 1$ , where:

$n$ - number of pentagonal, to which you want to add points to receive value of $n+1$ ;
$x$ - difference between $(n+1)^{\text{th}}$ and $n^{\text{th}}$ pentagonals.