3. 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
+ 9 + 15 + 21 + ...
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 series
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 from the first row.
ii. The upper limit of the sum is 18, as there are 18 rows in total.
iii. The explicit formula of the series is a_n = 6n - 3 where n is the row number.
b. To find the total number of beads in the necklace, we need to find the sum of the series. Since the explicit formula for the series is a_n = 6n - 3, we can calculate the sum using the formula for the sum of an arithmetic series:
S = n/2 * (first term + last term)
S = 18/2 * (3 + 6*18 - 3)
S = 9 * (3 + 108 - 3)
S = 9 * 108
S = 972
Therefore, the total number of beads in the necklace is 972.