PLEASE HELP ME ANSWER THIS QUESTION. THIS QUESTION IS HARD FOR ME AND i DON'T GET IT. please show working when answering. Just in case your wondering this is a practice question. Use the following pattern to answer the question
Design (n) Number of toothpicks(n)
1 5
2 9
3 13
4 17
5 21
6 25
Give the rule for the total number of toothpicks that Sarah would need if she was to continue following the pattern and complete ‘n’ designs.
Use this rule to find the total number of toothpicks needed to complete the first 12 designs using Sarah’s pattern.
Hints: I already know the equation for the nth design is 4n+1=t. But, the question is asking for the total equation.
To find the rule for the total number of toothpicks needed to complete 'n' designs, we need to analyze the given pattern.
Looking at the design number and the corresponding number of toothpicks, we can observe that the number of toothpicks increases by 4 for each design.
So, the equation for the number of toothpicks in the nth design can be written as:
Number of Toothpicks = 4n + 1
Now, to find the total number of toothpicks needed to complete 'n' designs, we need to sum up the toothpicks from the 1st design to the nth design.
To do this, we can use the sum formula for an arithmetic series:
Sum of an arithmetic series = (n/2) * (first term + last term)
In this case, the first term is the number of toothpicks in the 1st design (5) and the last term is the number of toothpicks in the nth design (4n + 1), which we derived earlier.
So, the total number of toothpicks needed to complete 'n' designs can be expressed as:
Total Number of Toothpicks = (n/2) * (5 + 4n + 1)
Simplifying this equation further, we get:
Total Number of Toothpicks = (n/2) * (4n + 6)
Now, let's find the total number of toothpicks needed to complete the first 12 designs.
Total Number of Toothpicks for n = 12:
Total Number of Toothpicks = (12/2) * (4(12) + 6)
= 6 * (48 + 6)
= 6 * 54
= 324
Therefore, the total number of toothpicks needed to complete the first 12 designs using Sarah's pattern is 324.