To determine the type of sequence, let's calculate the common ratio between consecutive terms:
The common ratio (r) can be found by dividing any term by its preceding term. Let's calculate it for the given sequence:
For the first pair of terms:
r = -32 / 64 = -0.5
For the second pair of terms:
r = 16 / -32 = -0.5
For the third pair of terms:
r = -8 / 16 = -0.5
For the fourth pair of terms:
r = 4 / -8 = -0.5
As we can see, the common ratio between consecutive terms is always -0.5. This indicates that the sequence is a geometric sequence.
Now, let's find the first term (a) in the sequence. By examining the pattern, we can see that the initial term is 64.
Hence, the type of sequence is geometric with a common ratio of -0.5, and the first term (a) is 64.
To find the sum of the series, we can use the formula for the sum of a geometric series:
S = a * (1 - r^n) / (1 - r)
Where:
S is the sum of the series
a is the first term
r is the common ratio
n is the number of terms
In this case, we have the following values:
a = 64
r = -0.5
n = 5 (since we have 5 terms in the sequence)
Let's substitute these values into the formula:
S = 64 * (1 - (-0.5)^5) / (1 - (-0.5))
Simplifying the equation:
S = 64 * (1 - 0.03125) / (1 + 0.5)
S = 64 * 0.96875 / 1.5
S = 61.875 / 1.5
S ≈ 41.25
Therefore, the sum of the series 64, -32, 16, -8, 4 is approximately 41.25.
The common ratio (r) can be found by dividing any term by its preceding term. Let's calculate it for the given sequence:
For the first pair of terms:
r = -32 / 64 = -0.5
For the second pair of terms:
r = 16 / -32 = -0.5
For the third pair of terms:
r = -8 / 16 = -0.5
For the fourth pair of terms:
r = 4 / -8 = -0.5
As we can see, the common ratio between consecutive terms is always -0.5. This indicates that the sequence is a geometric sequence.
Now, let's find the first term (a) in the sequence. By examining the pattern, we can see that the initial term is 64.
Hence, the type of sequence is geometric with a common ratio of -0.5, and the first term (a) is 64.
To find the sum of the series, we can use the formula for the sum of a geometric series:
S = a * (1 - r^n) / (1 - r)
Where:
S is the sum of the series
a is the first term
r is the common ratio
n is the number of terms
In this case, we have the following values:
a = 64
r = -0.5
n = 5 (since we have 5 terms in the sequence)
Let's substitute these values into the formula:
S = 64 * (1 - (-0.5)^5) / (1 - (-0.5))
Simplifying the equation:
S = 64 * (1 - 0.03125) / (1 + 0.5)
S = 64 * 0.96875 / 1.5
S = 61.875 / 1.5
S ≈ 41.25
Therefore, the sum of the series 64, -32, 16, -8, 4 is approximately 41.25.
To determine the number of rows in the pyramid, we can use the formula for the sum of an arithmetic series:
Sn = (n/2)(a + L)
Where:
Sn is the sum of the series
n is the number of terms
a is the first term
L is the last term
In this case, the first term (a) is 1, and the last term (L) is the number of cups in the topmost row, which is also 1. The sum of the series (Sn) is given as 820 cups.
Substituting these values into the formula:
820 = (n/2)(1 + 1)
820 = (n/2)(2)
820 = n
Therefore, there are 820 rows in the pyramid.
To find the total number of cups used in the pyramid, we can use the formula for the sum of an arithmetic series:
Sn = (n/2)(a + L)
In this case, the first term (a) is 1, the last term (L) is also 1, and the number of terms
is 820.Substituting these values into the formula:
Sn = (820/2)(1 + 1)
Sn = 410(2)
Sn = 820
Therefore, there are 820 cups used in the pyramid.