1. Summation notation:
- Lower limit of the sum: n = 1
- Upper limit of the sum: n = 18
- Explicit formula of the sum: Σ(3 + 7n), from n = 1 to 18.
2. Finding the total number of beads:
To find the total number of beads in the necklace, we need to calculate the sum of the series 3 + 10 + 17 + 24 + ... up to the 18th term.
First, we calculate the nth term of the series:
an = 3 + 7(n-1)
an = 3 + 7n - 7
an = 7n - 4
Now, calculate the sum of the series:
Σ(3 + 7n) = Σ 7n - Σ 4
Σ(7n) = 7Σn = 7(n)(n+1)/2
Σ(4) = 4(18) = 72
Substitute n = 18 into the formula:
7(18)(18+1)/2 - 72
7(18)(19)/2 - 72
7(342) - 72
2394 - 72
Total number of beads = 2322
Therefore, there are 2322 beads in Dante's necklace.
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.
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