An arithmetic series has a first term 8 and third term 20. What does the fifth term equal? What does the second partial sum equal?
The series is 8, 14, 20, 26, 32,...
The fifth term is 32.
The second partial sum is 8 + 14.
To find the fifth term of an arithmetic series, we need the common difference (d).
The common difference (d) can be found by subtracting the first term (a₁) from the second term (a₂), or by subtracting the second term (a₂) from the third term (a₃).
Given that the first term (a₁) is 8 and the third term (a₃) is 20, we can calculate the common difference (d) as follows:
d = a₃ - a₁
d = 20 - 8
d = 12
Now, we can find the fifth term (a₅) using the formula:
a₅ = a₁ + (n - 1) * d
Substituting the given values, we have:
a₅ = 8 + (5 - 1) * 12
a₅ = 8 + 4 * 12
a₅ = 8 + 48
a₅ = 56
So, the fifth term of the arithmetic series is 56.
Now, let's find the second partial sum (S₂). The second partial sum is the sum of the first two terms.
The formula to find the second partial sum is:
S₂ = (n/2) * (2a₁ + (n - 1) * d)
Substituting the given values, we have:
S₂ = (2/2) * (2 * 8 + (2 - 1) * 12)
S₂ = 1 * (16 + 12)
S₂ = 1 * 28
S₂ = 28
Therefore, the second partial sum of the arithmetic series is 28.
To find the fifth term of the arithmetic series, we need to identify the common difference (d) between each pair of consecutive terms.
The formula for an arithmetic series is:
An = A1 + (n - 1) * d
Where:
An represents the nth term of the series,
A1 is the first term,
n is the position of the term, and
d is the common difference.
Given that the first term (A1) is 8, we can find the common difference (d) by subtracting the second term (A2) from the first term (A1):
A2 = A1 + d
Since the second term is not known, we have to find it using the given information. We're told that the third term (A3) is 20. So, we can write:
A3 = A1 + 2d
Substituting the known values, we have:
20 = 8 + 2d
Simplifying, we find:
2d = 20 - 8
2d = 12
d = 6
Now, having found the common difference (d = 6), we can substitute it back into the original formula to find the fifth term (A5):
A5 = A1 + (n - 1) * d
A5 = 8 + (5 - 1) * 6
A5 = 8 + 4 * 6
A5 = 8 + 24
A5 = 32
Therefore, the fifth term of the arithmetic series is 32.
Moving on to the second partial sum, we can use the formula:
Sn = n/2 * (A1 + An)
Where:
Sn represents the sum of the first n terms of the series.
To find the second partial sum, we need to know the value of 'n' (the position of the term up to which we want to find the sum).
If we assume that 'n' is 2, we can calculate the second partial sum (S2):
S2 = 2/2 * (A1 + A2)
S2 = 1 * (8 + (8 + 6))
S2 = 1 * (8 + 14)
S2 = 1 * 22
S2 = 22
Therefore, the value of the second partial sum is 22.