Question
Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
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 3 + 10 + 17 + 24 + ... in summation notation, we can see that the pattern is increasing by 7 each time. So, the explicit formula for the series would be:
a_n = 3 + 7(n-1)
where n represents the position of the term in the series.
The lower limit of the sum would be n = 1 (the first term in the series), and the upper limit would be n = 18 (the 18th term in the series).
So, the summation notation for the series would be:
∑(3 + 7(n-1)), from n=1 to 18
To find the total number of beads in the necklace, we can calculate each term in the series and then sum them up:
a_1 = 3
a_2 = 3 + 7(2-1) = 10
a_3 = 3 + 7(3-1) = 17
...
a_18 = 3 + 7(18-1) = 3 + 7(17) = 3 + 119 = 122
Now, we can sum up all the terms from a_1 to a_18:
Total number of beads = 3 + 10 + 17 + ... + 122 = 18/2 * (3 + 122) = 9 * 125 = 1125
Therefore, the total number of beads in the necklace is 1125.
Are you sure?
I apologize for the mistake in my previous response.
Let's correct the calculation:
To find the total number of beads in the necklace, we need to find the sum of the series. The series is an arithmetic progression with a common difference of 7 and the first term is 3.
The nth term of an arithmetic progression is given by: a_n = a_1 + (n-1)d
where a_1 is the first term, d is the common difference, and n is the term number.
So, for this series, the nth term is: a_n = 3 + (n-1)7 = 3 + 7n - 7 = 7n - 4
To find the total number of beads, we need to find the sum of this series from n = 1 to n = 18:
Sum = (n/2)(first term + last term)
Sum = (18/2)(3 + 7*18 - 4)
Sum = 9(3 + 126 - 4)
Sum = 9(125)
Sum = 1125
Therefore, the correct total number of beads in the necklace is 1125.