Given:
The geometric sequence is
![4, -12, 36,...](https://tex.z-dn.net/?f=4%2C%20-12%2C%2036%2C...)
To find:
The sum of first 8 terms of the given geometric sequence.
Solution:
We have,
![4, -12, 36,...](https://tex.z-dn.net/?f=4%2C%20-12%2C%2036%2C...)
Here, the first term is 4 and the common ratio is
![r=\dfrac{-12}{4}](https://tex.z-dn.net/?f=r%3D%5Cdfrac%7B-12%7D%7B4%7D)
![r=-3](https://tex.z-dn.net/?f=r%3D-3)
The sum of first n terms of a geometric sequence is
![S_n=\dfrac{a(r^n-1)}{r-1}](https://tex.z-dn.net/?f=S_n%3D%5Cdfrac%7Ba%28r%5En-1%29%7D%7Br-1%7D)
Where, a is the first term and r is the common ratio.
Putting n=8, a=4 and r=-3, we get
![S_8=\dfrac{4((-3)^8-1)}{-3-1}](https://tex.z-dn.net/?f=S_8%3D%5Cdfrac%7B4%28%28-3%29%5E8-1%29%7D%7B-3-1%7D)
![S_8=\dfrac{4(6561-1)}{-4}](https://tex.z-dn.net/?f=S_8%3D%5Cdfrac%7B4%286561-1%29%7D%7B-4%7D)
![S_8=-6560](https://tex.z-dn.net/?f=S_8%3D-6560)
Therefore, the sum of first 8 terms is -6560.
Any variable that can have an infinite number of possible values will be considered continuous.
Let's compare the given function with the model for a quadratic equation:
![\begin{gathered} f(x)=ax^2+bx+b \\ a=2,b=12,c=-6 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20f%28x%29%3Dax%5E2%2Bbx%2Bb%20%5C%5C%20a%3D2%2Cb%3D12%2Cc%3D-6%20%5Cend%7Bgathered%7D)
Since the value of a is positive, the parabola has its concavity upwards, and the function has a minimum value.
The minimum value can be found calculating the y-coordinate of the vertex:
![\begin{gathered} x_v=-\frac{b}{2a}=-\frac{12}{4}=-3 \\ \\ y_v=2\cdot(-3)^2+12\cdot(-3)-6 \\ y_v=2\cdot9-36-6^{} \\ y_v=-24 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20x_v%3D-%5Cfrac%7Bb%7D%7B2a%7D%3D-%5Cfrac%7B12%7D%7B4%7D%3D-3%20%5C%5C%20%20%5C%5C%20y_v%3D2%5Ccdot%28-3%29%5E2%2B12%5Ccdot%28-3%29-6%20%5C%5C%20y_v%3D2%5Ccdot9-36-6%5E%7B%7D%20%5C%5C%20y_v%3D-24%20%5Cend%7Bgathered%7D)
Therefore the minimum value is -24.
Answer:
C
Step-by-step explanation:
It would be c i took the test:)