This sequence has generating function
![F(x)=\displaystyle\sum_{k\ge0}k^3x^k](https://tex.z-dn.net/?f=F%28x%29%3D%5Cdisplaystyle%5Csum_%7Bk%5Cge0%7Dk%5E3x%5Ek)
(if we include
for a moment)
Recall that for
, we have
![\displaystyle\frac1{1-x}=\sum_{k\ge0}x^k](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cfrac1%7B1-x%7D%3D%5Csum_%7Bk%5Cge0%7Dx%5Ek)
Take the derivative to get
![\displaystyle\frac1{(1-x)^2}=\sum_{k\ge0}kx^{k-1}=\frac1x\sum_{k\ge0}kx^k](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cfrac1%7B%281-x%29%5E2%7D%3D%5Csum_%7Bk%5Cge0%7Dkx%5E%7Bk-1%7D%3D%5Cfrac1x%5Csum_%7Bk%5Cge0%7Dkx%5Ek)
![\implies\dfrac x{(1-x)^2}=\displaystyle\sum_{k\ge0}kx^k](https://tex.z-dn.net/?f=%5Cimplies%5Cdfrac%20x%7B%281-x%29%5E2%7D%3D%5Cdisplaystyle%5Csum_%7Bk%5Cge0%7Dkx%5Ek)
Take the derivative again:
![\displaystyle\frac{(1-x)^2+2x(1-x)}{(1-x)^4}=\sum_{k\ge0}k^2x^{k-1}=\frac1x\sum_{k\ge0}k^2x^k](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cfrac%7B%281-x%29%5E2%2B2x%281-x%29%7D%7B%281-x%29%5E4%7D%3D%5Csum_%7Bk%5Cge0%7Dk%5E2x%5E%7Bk-1%7D%3D%5Cfrac1x%5Csum_%7Bk%5Cge0%7Dk%5E2x%5Ek)
![\implies\displaystyle\frac{x+x^2}{(1-x)^3}=\sum_{k\ge0}k^2x^k](https://tex.z-dn.net/?f=%5Cimplies%5Cdisplaystyle%5Cfrac%7Bx%2Bx%5E2%7D%7B%281-x%29%5E3%7D%3D%5Csum_%7Bk%5Cge0%7Dk%5E2x%5Ek)
Take the derivative one more time:
![\displaystyle\frac{(1+2x)(1-x)^3+3(x+x^2)(1-x)^2}{(1-x)^6}=\sum_{k\ge0}k^3x^{k-1}=\frac1x\sum_{k\ge0}k^3x^k](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cfrac%7B%281%2B2x%29%281-x%29%5E3%2B3%28x%2Bx%5E2%29%281-x%29%5E2%7D%7B%281-x%29%5E6%7D%3D%5Csum_%7Bk%5Cge0%7Dk%5E3x%5E%7Bk-1%7D%3D%5Cfrac1x%5Csum_%7Bk%5Cge0%7Dk%5E3x%5Ek)
![\implies\displaystyle\frac{x+4x^3+x^3}{(1-x)^4}=\sum_{k\ge0}k^3x^k](https://tex.z-dn.net/?f=%5Cimplies%5Cdisplaystyle%5Cfrac%7Bx%2B4x%5E3%2Bx%5E3%7D%7B%281-x%29%5E4%7D%3D%5Csum_%7Bk%5Cge0%7Dk%5E3x%5Ek)
so we have
![\boxed{F(x)=\dfrac{x+4x^3+x^3}{(1-x)^4}}](https://tex.z-dn.net/?f=%5Cboxed%7BF%28x%29%3D%5Cdfrac%7Bx%2B4x%5E3%2Bx%5E3%7D%7B%281-x%29%5E4%7D%7D)
For #1, C. 1080, For #2, B. 7, For #10, I think its because of the reflexive property (honestly, Google it just to be safe) , And for #11, A Rhombus, Having trouble remembering why but I know its not a rectangle or square.
Answer:
3200000
Step-by-step explanation:
multiply the volume value by 1000
The standard deviation is the square root of the variance
so if the variance is 3.6, the standard deviation is the sqrt 3.6...
sqrt 3.6 = 1.897 <= if u need it rounded, it is 1.9 grams