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+...
a. If you were to write this series in summation notation, give
i. the lower limit of the sum
ii. the upper limit of the sum
iii. the explicit formula of the sum
b. Find the total number of beads in the necklace. Explain your method for finding the total number of beads.
Show all your work.
Thank you!
a. To write the series in summation notation, we need to find the general term of the series.
The general term is given by the formula Tn = a + (n-1)d, where Tn is the nth term, a is the first term, n is the number of terms, and d is the common difference.
In this case, the first term (a) is 3, and the common difference (d) is 7 (10 - 3).
i. The lower limit of the sum is the first term, so it is a = 3.
ii. The upper limit of the sum is the last term, which we need to find.
We can find it by using the formula for the nth term:
Tn = a + (n-1)d
Plugging in the values, we get:
24 = 3 + (n-1)7
24 - 3 = 7n - 7
21 = 7n - 7
28 = 7n
n = 4
So the upper limit of the sum is n = 4.
iii. The explicit formula of the sum is ∑(k=1 to 4) (3 + (k-1)7), where k represents each term from 1 to 4.
b. To find the total number of beads in the necklace, we need to find the sum of the series. We can use the summation formula for the arithmetic series.
The sum (S) of an arithmetic series is given by the formula:
S = (n/2)(a + L), where n is the number of terms, a is the first term, and L is the last term.
Plugging in the values, we have:
n = 4 (from part a)
a = 3 (given)
L = 3 + (4-1)7 = 3 + 3(7) = 3 + 21 = 24 (from part a)
So, the total number of beads in the necklace is:
S = (4/2)(3 + 24) = 2(27) = 54
Therefore, the necklace has a total of 54 beads.
a. To write the given series in summation notation, we need to determine the following:
i. The lower limit of the sum: The lower limit of the sum is usually denoted by the index variable. In this case, it represents the starting position of the series. Looking at the series, we can see that it starts with the term "3." Therefore, the lower limit of the sum is "n = 1."
ii. The upper limit of the sum: The upper limit of the sum represents the ending position of the series. In this case, the series is given in a pattern, and we need to find the number of terms (rows of tiny beads). Each row adds 7 more beads than the previous one (3, 10, 17, 24, ...), thus forming an arithmetic progression with a common difference of 7. To find the upper limit, we need to determine the number of terms in the progression. Since the series starts with 3 and increases by 7 each time, we can write the nth term as: aₙ = 3 + (n - 1) * 7. The nth term represents the number of beads in the nth row.
Using the formula for the nth term, we can find the number of terms (rows) in the series:
aₙ = 3 + (n - 1) * 7
18 = 3 + (n - 1) * 7
15 = (n - 1) * 7
n - 1 = 15 / 7
n - 1 ≈ 2.14
Since the number of terms (rows) needs to be a whole number, we round up to the nearest whole number.
n ≈ 3
Therefore, the upper limit of the sum is "n = 3."
iii. The explicit formula of the sum: The series can be written as Σ(3 + 7(n - 1)), with "n" going from 1 to 3. This notation represents the sum of terms in the series from the lower limit to the upper limit.
b. Now, let's find the total number of beads in the necklace using the explicit formula of the sum.
The explicit formula states: Σ(3 + 7(n-1)), from n = 1 to 3.
To evaluate the sum, we substitute "n" with the values from the lower to the upper limit and sum the terms:
n = 1: 3 + 7(1 - 1) = 3
n = 2: 3 + 7(2 - 1) = 10
n = 3: 3 + 7(3 - 1) = 17
Now, we sum the terms:
3 + 10 + 17 = 30
Therefore, the total number of beads in the necklace is 30 beads.
note that each term increases by 7
So, you have an AP with
a = 3
d = 7
So, S_18 is
18
∑ 3+7(n-1) = 18/2 (2*3 + 17*7)
n=1