Determine whether the finite series is arithmetic, geometric, both, or neither. If the series is arithmetic or geometric, find its sum.
20+19.6+19.2+...+0.4
The series is arithmetic because there is a common difference between each term. The common difference is -0.4.
To find the sum of an arithmetic series, we can use the formula:
Sn = n/2(2a + (n-1)d)
Where:
Sn = sum of the first n terms
n = number of terms
a = first term
d = common difference
In this series, a = 20, d = -0.4, and we need to find n.
To find n, we can use the formula for the nth term of an arithmetic series:
an = a + (n-1)d
In this series, we need to find the value of n when an = 0.4, a = 20, and d = -0.4.
0.4 = 20 + (n-1)(-0.4)
0.4 = 20 - 0.4n + 0.4
0.4n = 20
n = 50
Now that we have the value of n, we can find the sum of the series:
Sn = 50/2(2(20) + (50-1)(-0.4))
Sn = 25(40 + 49(-0.4))
Sn = 400 + 25(-19.6)
Sn = 400 - 490
Sn = -90
Therefore, the sum of the series 20+19.6+19.2+...+0.4 is -90.
To determine whether the given series is arithmetic, geometric, both, or neither, we need to check if the difference between consecutive terms is constant (for arithmetic) or if the ratio between consecutive terms is constant (for geometric).
To do this, let's find the common difference or common ratio.
Common difference (d) = 19.6 - 20 = -0.4
Common ratio (r) = 19.6 / 20 = 0.98
Since the common difference (-0.4) is constant, the given series is arithmetic.
To find the sum of an arithmetic series, we can use the formula:
Sn = (n/2)(a + l)
where Sn is the sum of the series, n is the number of terms, a is the first term, and l is the last term.
In this series, the first term (a) is 20, and the last term (l) is 0.4. Since we want to find the sum up to and including the 0.4 term, we can write l as the nth term.
The nth term (an) can be found using the formula:
an = a + (n-1)d
where d is the common difference (-0.4).
For the nth term to be 0.4:
0.4 = 20 + (n-1)(-0.4)
Simplifying the equation:
0.4 = 20 - 0.4n + 0.4
0.4n = 20
n = 20 / 0.4
n = 50
So, the number of terms (n) in the series is 50.
Now we can calculate the sum (Sn):
Sn = (n/2)(a + l)
= (50/2)(20 + 0.4)
= 25 * 20.4
= 510
Therefore, the sum of the given finite arithmetic series is 510.
To determine if the given finite series is arithmetic, geometric, both, or neither, we need to examine the common difference or common ratio between the consecutive terms.
First, let's calculate the common difference between each term. Subtracting each term from its previous term, we have:
19.6 - 20 = -0.4
19.2 - 19.6 = -0.4
18.8 - 19.2 = -0.4
...
Since the common difference between each term is constant (-0.4), this series is an arithmetic series.
To find the sum of an arithmetic series, we can use the following formula:
S = (n/2)(a + l)
Where:
S is the sum of the series.
n is the number of terms.
a is the first term.
l is the last term.
The first term a is 20, and the common difference d is -0.4. To find the last term, we need to determine when the last term becomes 0.4:
20 + (n-1)*(-0.4) = 0.4
Simplifying this equation, we get:
-0.4n = -19.6
Dividing both sides by -0.4, we find that n = 49.
Now, we can calculate the sum S:
S = (49/2)(20 + 0.4)
S = 24.7(20.4)
S = 504.18
Therefore, the sum of the given arithmetic series is 504.18.