MHB How to Calculate Total Toothpicks and Squares in Any Figure of the Sequence?

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To calculate the total number of toothpicks and squares in a sequence of figures, a pattern emerges from the differences in counts. The number of toothpicks increases by an even number sequence starting from 4, while the number of squares follows a triangular number pattern. A formula can be derived for the number of toothpicks using the arithmetic progression of the increases, with the first term as 4 and a common difference of 2. The challenge lies in formulating a general expression for any figure number without manually listing each figure. Understanding these patterns is crucial for developing a formula that accurately predicts the counts for any figure in the sequence.
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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!
 

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marsman said:
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?
 
Opalg said:
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.
[table="width: 500, class: grid"]
[tr][td]Figure Number[/td]
[td]1[/td]
[td]2[/td]
[td]3[/td]
[td]4[/td]
[td]5[/td]
[/tr]
[tr][td]Number of Toothpicks[/td]
[td]4[/td]
[td]10[/td]
[td]18[/td]
[td]28[/td]
[td]40[/td]
[/tr]
[tr][td]Increase in Toothpicks[/td]
[td]+4[/td]
[td]+6[/td]
[td]+8[/td]
[td]+10[/td]
[td]+12[/td]
[/tr]
[tr][td]Number of Squares[/td]
[td]1[/td]
[td]3[/td]
[td]6[/td]
[td]10[/td]
[td]15[/td]
[/tr]
[tr][td]Increase in Squares[/td]
[td]+1[/td]
[td]+2[/td]
[td]+3[/td]
[td]+4[/td]
[td]+5[/td]
[/tr]
[/table]
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
 
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?
 
marsman said:
Yeah I'm up to my neck in patterns; I made a table and everything.
[TABLE="class: grid, width: 500"]
[TR]
[TD]Figure Number[/TD]
[TD]1[/TD]
[TD]2[/TD]
[TD]3[/TD]
[TD]4[/TD]
[TD]5[/TD]
[/TR]
[TR]
[TD]Number of Toothpicks[/TD]
[TD]4[/TD]
[TD]10[/TD]
[TD]18[/TD]
[TD]28[/TD]
[TD]40[/TD]
[/TR]
[TR]
[TD]Increase in Toothpicks[/TD]
[TD]+4[/TD]
[TD]+6[/TD]
[TD]+8[/TD]
[TD]+10[/TD]
[TD]+12[/TD]
[/TR]
[TR]
[TD]Number of Squares[/TD]
[TD]1[/TD]
[TD]3[/TD]
[TD]6[/TD]
[TD]10[/TD]
[TD]15[/TD]
[/TR]
[TR]
[TD]Increase in Squares[/TD]
[TD]+1[/TD]
[TD]+2[/TD]
[TD]+3[/TD]
[TD]+4[/TD]
[TD]+5[/TD]
[/TR]
[/TABLE]
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 $x_n$?

[sp]Hints: arithmetic progression, $n$ terms, first term $4$, common difference $2$.[/sp]
 
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