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Question

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.

Answer 1

To write the series in summation notation, we can see that it is an arithmetic series where the common difference is 7 and the first term is 3.

So, the series can be written as: 3 + 10 + 17 + 24 + ... = Σ(3 + 7(n-1)), from n=1 to 18

Now we can find the explicit formula of the sum:
Σ(3 + 7(n-1)) = Σ3 + Σ7(n-1) = 3(18) + 7(Σ(n-1))

Σ(n-1) from 1 to 18 is actually the sum of integers from 0 to 17, which is a well-known arithmetic series. The formula for the sum of the first n non-negative integers is n(n+1)/2, so in this case, it would be 17(17+1)/2 = 17(18)/2 = 17*9 = 153.

Now we substitute this back into our formula to get:
3(18) + 7(153) = 54 + 1071 = 1125

Therefore, the total number of beads in the necklace is 1125.