Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
Dante is making a necklace with 18 rows of tiny beads in which the number of beads per row is given by the series 3 + 10 + 17 + 24 + ...
If you were to write this series in summation notation, give
the lower limit of the sum
the upper limit of the sum
the explicit formula of the sum
Find the total number of beads in the necklace. Explain your method for finding the total number of beads.
Show all your work.
To find the explicit formula of the series, we can see that it is an arithmetic series with a common difference of 7 starting from 3. So, the explicit formula of the series can be written as:
an = 3 + 7(n-1) = 7n - 4
To find the lower limit, we need to find the value of n when the series starts. Given that the first term is 3, we can plug in n=1 into the explicit formula:
a1 = 7(1) - 4 = 3
Therefore, the lower limit of the sum is 1.
To find the upper limit, we need to find the value of n when the series ends. Given that there are 18 rows in the necklace, we can plug in n=18 into the explicit formula:
a18 = 7(18) - 4 = 122
Therefore, the upper limit of the sum is 18.
To find the total number of beads in the necklace, we can use the formula for the sum of an arithmetic series:
S = n/2 * (a1 + an)
S = 18/2 * (3 + 122)
S = 9 * 125
S = 1125
Therefore, the total number of beads in the necklace is 1125.