# Toothpick Squares Sequence

#### marsman

##### New member
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.
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
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
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 Number 1 2 3 4 5 Number of Toothpicks 4 10 18 28 40 Increase in Toothpicks +4 +6 +8 +10 +12 Number of Squares 1 3 6 10 15 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

Staff member
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
Yeah I'm up to my neck in patterns; I made a table and everything.
 Figure Number 1 2 3 4 5 Number of Toothpicks 4 10 18 28 40 Increase in Toothpicks +4 +6 +8 +10 +12 Number of Squares 1 3 6 10 15 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 $2$.