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 + ...
a. If you were to write this series in summation notation, give
i.the lower limit of the sum
ii.the upper limit of the sum
iii.the explicit formula of the sum
b. Find the total number of beads in the necklace. Explain your method for finding the total number of beads.
a.
i. The lower limit of the sum is 1, as the series starts with the first row having 3 beads.
ii. The upper limit of the sum is 18, as there are 18 rows in the necklace.
iii. The explicit formula of the sum is: S = Σ(3 + 7n) from n=1 to 18
b. To find the total number of beads in the necklace, we need to sum up the total number of beads in each row. To do this, we can use the formula for the sum of an arithmetic series:
S = n/2 * (first term + last term) where n is the number of terms
Using this formula, we can calculate the total number of beads in the necklace as follows:
S = 18/2 * (3 + (3 + 7 * 17))
S = 9 * (3 + 122)
S = 9 * 125
S = 1125
So, the total number of beads in the necklace is 1125.