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.
To write the series in summation notation, we notice that the pattern of the series is an arithmetic sequence with a common difference of 7 starting from 3. Therefore, the explicit formula for the number of beads in the nth row is given by:
a(n) = 3 + 7(n-1)
The lower limit of the sum is n=1 (since we start from the first row) and the upper limit of the sum is n=18 (since there are 18 rows in total).
Therefore, the series can be written in summation notation as:
Σ[3 + 7(n-1)] from n=1 to 18
To find the total number of beads in the necklace, we need to find the sum of the series. Using the formula for the sum of an arithmetic series, which is given by:
S = n/2 [2a + (n-1)d]
where S is the sum, n is the number of terms, a is the first term, and d is the common difference, we can plug in the values:
n = 18 (number of rows)
a = 3 (first term)
d = 7 (common difference)
S = 18/2 [2(3) + (18-1)7]
S = 9 [6 + 119]
S = 9 [125]
S = 1125
Therefore, the total number of beads in the necklace is 1125.