MHB How Do You Determine the Number of Toothpicks in a Stack of Squares?

[Gurtejtaggar]

I am doing a problem in math which I can't seem to solve.The problem is apart of my discrete functions unit it grade 11 math. The question is: Toothpicks are used to make a sequence of stacked squares as shown. Determine a rule for calculating the total number of toothpicks needed for a stack of squares that is n high. It also gave a diagram with 4 terms and the number of toothpicks in each term was 4,10,18,28. I know that the first differences are 6,8 and 10 which means the second differences share a constant value of 2. Does this mean that I need to make a quadratic equation to find the answer or something else? If so, how would I? By the way, this problem is a thinking or extension problem.

 

[MarkFL]

Your observations are correct...we can represent the total $T$ number of toothpicks for a stack $n$ levels high with the function:

$$T(n)=k_1n^2+k_2n+k_3$$

So, what you want to do is use the given values to determine the values of the parameters $k_i$.

$$T(1)=k_1+k_2+k_3=4$$

$$T(2)=4k_1+2k_2+k_3=10$$

$$T(3)=9k_1+3k_2+k_3=18$$

So, you have a linear 3X3 system to solve.

By the way, I am going to delete the duplicate of this thread posted in our "Chat Room" forum. :D

 

[MarkFL]

We could simplify matters by observing $T(0)=0$ and since then $k_3=0$ reduce the system to:

$$k_1+k_2=4$$

$$2k_1+k_2=5$$

Subtracting the former from the latter, we obtain:

$$k_1=1\implies k_2=3$$

And so we have:

$$T(n)=n^2+3n=n(n+3)$$