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 + ...
1.. If you were to write this series in summation notation, give the lower limit of the sum, the upper limit of the sum, and the explicit formula of the sum.
2. Find the total number of beads in the necklace. Explain your method for finding the total number of beads.
1. The series given can be written as: Σ(3 + 7n) from n=0 to 17.
Lower limit of the sum: n=0
Upper limit of the sum: n=17
Explicit formula of the sum: Σ(3 + 7n) = 3(18) + 7Σn = 54 + 7(0+1+2+...+17) = 54 + 7(17)(18)/2 = 648.
2. To find the total number of beads in the necklace, we can use the formula for the sum of an arithmetic series: Sn = n/2 * (a1 + an), where Sn is the sum of the series, n is the number of terms, a1 is the first term of the series, and an is the last term of the series.
In this case, n = 18 (number of rows), a1 = 3 (number of beads in the first row), and an = 24 (number of beads in the last row).
Total number of beads = 18/2 * (3 + 24) = 9 * 27 = 243.
Therefore, the necklace will have a total of 243 beads.