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. Use summation notation to write the series, give
i.the lower limit of the sum
ii.the upper limit of the sum
iii.the explicit formula of the sum
Please help? Thanks
To find the summation notation for the given series, we need to identify the pattern and properties of the series.
The series 3, 10, 17, 24, ... follows an arithmetic progression with a common difference of 7. We can observe that each term is obtained by adding 7 to the previous term, starting from the first term of 3.
Now, let's break down the components of the summation notation:
i. Lower limit of the sum:
The lower limit represents the starting point of the series. In this case, the series starts with the first term of 3. So, the lower limit of the sum is 1.
ii. Upper limit of the sum:
The upper limit specifies the ending point of the series. Since the series has 18 rows, the upper limit of the sum is 18.
iii. Explicit formula of the sum:
To find the explicit formula of the sum, we can use the formula for the sum of an arithmetic series:
Sum = (n/2)(first term + last term),
where n is the number of terms in the series.
First, let's find the last term:
Last term = first term + (n - 1) * common difference
= 3 + (18 - 1) * 7
= 3 + 17 * 7
= 3 + 119
= 122.
Now, we can substitute the values into the explicit formula:
Sum = (18/2)(3 + 122)
= 9(125)
= 1125.
Therefore, the summation notation for the series is:
Σ(i = 1 to 18)(3 + (i - 1) * 7),
where i represents the row number and the sum gives the total number of beads in the necklace.
To write the series using summation notation, we need to find the pattern and the formula for the series. The series starts with 3 and each subsequent term increases by 7.
a. Finding the pattern:
3, 10 (3+7), 17 (10+7), 24 (17+7), ...
b. Finding the formula for the nth term:
The general formula for an arithmetic sequence is given by: an = a1 + (n - 1)d, where "an" represents the nth term, "a1" is the first term, and "d" is the common difference.
In this case, the first term (a1) is 3, and the common difference (d) is 7. Therefore, the general formula for the nth term is: an = 3 + (n - 1)7.
c. Writing the series using summation notation:
To write the series using summation notation, we use the Greek letter sigma (Σ).
i. The lower limit of the sum: We need to find the value of n for the first term of the series. In this case, the first term is 3, so the lower limit of the sum is n = 1.
ii. The upper limit of the sum: We need to find the value of n for the last term of the series. Looking at the pattern, we can see that each term increases by 7. So, for the last term, let's set an = 18. Solving for n, we get:
18 = 3 + (n - 1)7
18 = 3 + 7n - 7
22 = 7n
n = 22/7
n ≈ 3.1429
Since n represents the number of rows, we can round up to the nearest whole number (since we can't have a fraction of a row). Therefore, the upper limit of the sum is n = 4.
iii. The explicit formula of the sum: We can use the sigma notation to express the given series as:
Σ (3 + 7(n-1)), where n starts at 1 and goes up to 4.
Thus, the summation notation for the series is:
Σ (3 + 7(n-1)), n = 1 to 4.
17
∑ 3+7k
k=0
S18 = 18/2 (2*3+17*7)