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.
The series given is an arithmetic progression with first term a = 3 and common difference d = 7.
The sum of the first n terms of an arithmetic progression can be calculated using the formula:
S_n = n/2 * (2a + (n-1)d)
In this case, we are considering 18 rows of beads, so n = 18.
The lower limit of the sum is 1 (for the first term of the series)
The upper limit of the sum is 18
So, the explicit formula of the sum is:
S_18 = 18/2 * (2*3 + (18-1)7)
= 9 * (6 + 17*7)
= 9 * (6 + 119)
= 9 * 125
= 1125
Therefore, the total number of beads in the necklace is 1125.
To find the total number of beads, we calculated the sum of the first 18 terms of the series by using the formula for the sum of an arithmetic progression.