Respuesta :
Answer: −59.9356.
Step-by-step explanation:
To find the sum of the given series, we can write it in sigma notation:
∑
�
=
1
�
20
⋅
3
�
3
�
∑
n=1
n
3
n
20⋅3
n
This is a geometric series with the first term
�
=
20
a=20 and the common ratio
�
=
4
3
r=
3
4
(because each term is obtained by multiplying the previous term by
4
3
3
4
).
The sum of a geometric series is given by the formula:
�
=
�
(
1
−
�
�
)
1
−
�
S=
1−r
a(1−r
n
)
where:
�
S is the sum of the series
�
a is the first term
�
r is the common ratio
�
n is the number of terms
We need to find the value of
�
n such that
�
�
=
10240
19683
r
n
=
19683
10240
, then we can plug in the values into the formula to find the sum.
4
�
3
�
=
10240
19683
3
n
4
n
=
19683
10240
(
4
3
)
�
=
10240
19683
(
3
4
)
n
=
19683
10240
(
4
3
)
�
=
(
2
10
3
10
)
⋅
(
1
3
6
)
(
3
4
)
n
=(
3
10
2
10
)⋅(
3
6
1
)
(
4
3
)
�
=
(
2
3
)
10
⋅
(
1
3
6
)
(
3
4
)
n
=(
3
2
)
10
⋅(
3
6
1
)
(
4
3
)
�
=
(
2
3
)
16
(
3
4
)
n
=(
3
2
)
16
Comparing with
(
4
3
)
�
=
10240
19683
(
3
4
)
n
=
19683
10240
, we have
�
=
16
n=16.
Now, let's plug in the values into the formula:
�
=
20
(
1
−
(
4
/
3
)
16
)
1
−
4
/
3
S=
1−4/3
20(1−(4/3)
16
)
Calculating this expression gives us:
�
=
20
(
1
−
(
2
/
3
)
16
)
−
1
/
3
S=
−1/3
20(1−(2/3)
16
)
�
=
−
60
(
1
−
(
2
/
3
)
16
)
S=−60(1−(2/3)
16
)
Now, let's compute
(
2
/
3
)
16
(2/3)
16
:
(
2
/
3
)
16
≈
0.0006067817
(2/3)
16
≈0.0006067817
So,
�
=
−
60
(
1
−
0.0006067817
)
S=−60(1−0.0006067817)
�
≈
−
60
(
0.9993932183
)
S≈−60(0.9993932183)
�
≈
−
59.93559251
S≈−59.93559251
Rounded to four decimal places, the sum of the series is approximately
−
59.9356
−59.9356.
