Answer:
[tex] - 13107[/tex]
Step-by-step explanation:
We would like to evaluate the following Geometric Series ,
[tex]\displaystyle\longrightarrow \sum _{n=1}^8 (-4)^{n-1 }[/tex]
In a geometric series, a common number is multiplied to the previous term in order to find the next term . And that common number is called common ratio (r) .
As we know that ,
[tex]\displaystyle\longrightarrow \sum_{x = 1}^n f(x) = f(1) + f(2) + \dots + f(n) [/tex]
So , we can write the series as ,
[tex]\displaystyle\longrightarrow (-4)^{1-1}+(-4)^{2-1}+\dots +(-4)^{8-1} [/tex]
Simplify,
[tex]\displaystyle\longrightarrow (-4)^0 + (-4)^1+(-4)^2+\dots +(-4)^7 [/tex]
We can find the common ratio by dividing and successive term by its preceding term , as ;
[tex]\displaystyle\longrightarrow r =\dfrac{-4}{1}=-4 [/tex]
Again , here ;
- First term = a = [tex](-4)^0=1[/tex]
- Common ratio = [tex]-4[/tex]
- number of terms (n) = 8
And we can find the sum of geometric series using the formula , ( this is used when the value of r is less than 1 , here it is -4 ) .
[tex]\displaystyle\longrightarrow \Bigg[ Sum = \dfrac{a(1-r^n)}{1-r}\Bigg] [/tex]
- where the symbols have their usual meaning.
On substituting the respective values, we have;
[tex]\displaystyle \longrightarrow\rm{ Sum }=\dfrac{ 1\{1-(-4)^8\}}{1-(-4)}\\ [/tex]
Simplify ,
[tex]\displaystyle\longrightarrow \rm{Sum} = \dfrac{1-65536}{1+4}\\[/tex]
[tex]\displaystyle\longrightarrow \rm{Sum} = \dfrac{-65535}{5} [/tex]
Simplify by dividing ,
[tex]\displaystyle\longrightarrow \underline{\underline{ \rm{Sum} = -13107}} [/tex]
And we are done !
