Answer:
Z=1
Step-by-step explanation:
This is a geometric progression
7-14Z+28Z^2-56Z^3
a=7
r=2
Sn=a(r^n-1)/(r-1)
Sn=7(2^n-1)/1
Sn=7(2^n-1)
Sinfinity=a/(1-r)
Sinfinity=7
7(2^n-1)=7
2^n-1=1
2^n=2
n=1
The series is an illustration of a geometric series.
The series is given as: [tex]\mathbf{7 - 14z+28z^2 -56z^3+...}[/tex]
So, we have:
[tex]\mathbf{a = 7}[/tex] -- first term
[tex]\mathbf{r= -2z}[/tex] --- common ratio
The sum of n terms of a geometric series is:
[tex]\mathbf{S_n = \frac{a(r^n - 1)}{r - 1}}[/tex]
The number of elements in the series is 4.
So, we have:
[tex]\mathbf{S_4 = \frac{7((-2z)^4 - 1)}{-2z - 1}}[/tex]
[tex]\mathbf{S_4 = \frac{7(16z^4 - 1)}{-2z - 1}}[/tex]
[tex]\mathbf{S_4 = -\frac{7(16z^4 - 1)}{2z + 1}}[/tex]
Hence, the sum of the elements in the series is [tex]\mathbf{-\frac{7(16z^4 - 1)}{2z + 1}}[/tex]
When the sum converges, we have:
[tex]\mathbf{S_{\infty} = \frac{a}{1 -r}}[/tex]
So, we have:
[tex]\mathbf{S_{\infty} = \frac{7}{1 +2z}}[/tex]
The sum converges at S4.
So, we have:
[tex]\mathbf{\frac{7}{1 +2z} = -\frac{7(16z^4 - 1)}{2z + 1}}[/tex]
Cancel out common factors
[tex]\mathbf{1 = -(16z^4 - 1)}[/tex]
Divide both sides by -1
[tex]\mathbf{-1 = 16z^4 - 1}[/tex]
Add 1 to both sides
[tex]\mathbf{0 = 16z^4 }[/tex]
Divide by 16
[tex]\mathbf{0 = z^4 }[/tex]
Take 4th roots
[tex]\mathbf{0 = z}[/tex]
Rewrite as:
[tex]\mathbf{z = 0}[/tex]
Hence, the variable is 0, when the sum converges at [tex]\mathbf{-\frac{7(16z^4 - 1)}{2z + 1}}[/tex]
Read more about geometric series at:
https://brainly.com/question/4617980
