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.
The series given is an arithmetic series with a first term of 3 and a common difference of 7. The nth term of the series is given by the formula a_n = 3 + 7(n-1).
Writing the series in summation notation:
∑(3 + 7(n-1)), from n=1 to 18
Lower limit of the sum: 1
Upper limit of the sum: 18
The explicit formula of the sum is:
∑(3 + 7(n-1)) = 3(18) + 7(1+2+...+17) = 54 + 7(153) = 54 + 1071 = 1125
Therefore, the total number of beads in the necklace is 1125.
The method for finding the total number of beads was to calculate the sum of the series using the explicit formula of the sum for an arithmetic series.