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Toothpick Squares Sequence

marsman

New member
Oct 29, 2013
2
Please help! I need to be able to find the number of total toothpicks used in any given figure in this sequence, in addition to total squares in any figure, using the figure number.fig number stacked.png
The only way I can figure this out is by listing them all out one by one, which is TERRIBLE :(
Thank you so much in advance!!!!!
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,707
Please help! I need to be able to find the number of total toothpicks used in any given figure in this sequence, in addition to total squares in any figure, using the figure number.View attachment 1592
The only way I can figure this out is by listing them all out one by one, which is TERRIBLE :(
Thank you so much in advance!!!!!
Suggestion: In the six figures, count how many toothpicks you have to add to each figure in order to get the next one. Can you see a pattern there?
 

marsman

New member
Oct 29, 2013
2
Suggestion: In the six figures, count how many toothpicks you have to add to each figure in order to get the next one. Can you see a pattern there?
Yeah I'm up to my neck in patterns; I made a table and everything.
Figure Number12345
Number of Toothpicks410182840
Increase in Toothpicks+4+6+8+10+12
Number of Squares1361015
Increase in Squares+1+2+3+4+5
My problem is that I can't get from the patterns to creating a formula that would work for any figure number without listing them all out consecutively
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
I see you have found that the difference in the number of toothpicks per iteration (where you state the increase in toothpicks) are the sequence of even numbers, beginning with 4. What is the second difference, that is, the difference of the differences. How much does the increase increase each time?
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,707
Yeah I'm up to my neck in patterns; I made a table and everything.
Figure Number12345
Number of Toothpicks410182840
Increase in Toothpicks+4+6+8+10+12
Number of Squares1361015
Increase in Squares+1+2+3+4+5
My problem is that I can't get from the patterns to creating a formula that would work for any figure number without listing them all out consecutively
Let $x_n$ be the number of toothpicks in Fig. $n$. Your table shows that $x_1=4$, $x_2=4+6$, $x_3=4+6+8$, $x_4=4+6+8+10$, $\ldots$. So what is the formula for $x_n$?

Hints: arithmetic progression, $n$ terms, first term $4$, common difference $2$.