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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 find the total number of beads in the necklace, we need to determine the number of beads in each row and then sum them up for all 18 rows.
The series given is: 3 + 10 + 17 + 24 + ...
This is an arithmetic series with a common difference of 7 between each term.
To find the nth term of the series, we can use the formula an = a1 + (n-1)d, where an is the nth term, a1 is the first term, n is the term number, and d is the common difference.
So, the nth term of the series is an = 3 + (n-1)7 = 3 + 7n - 7 = 7n - 4.
Now, we need to find the total number of beads in the necklace by summing up the terms of the series up to the 18th row. This can be represented in summation notation as:
∑(n=1 to 18) (7n - 4)
Now, we need to find the lower limit, upper limit, and the explicit formula of the sum.
Lower limit: 1 (as we start at the 1st term)
Upper limit: 18
Explicit formula of the sum: Σ(7n - 4) from n=1 to 18
To find the total number of beads, we can now sum up the terms from 1 to 18 using the explicit formula:
Σ(7n - 4) from n=1 to 18
= 7(1) - 4 + 7(2) - 4 + 7(3) - 4 + ... + 7(18) - 4
= 7 + 10 + 13 + ... + 122
= 2164
Therefore, the total number of beads in the necklace is 2164.