<h3>
Answer: 360</h3>
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Explanation:
We have 3 even values (2,4, and 6) so this is the number of choices we have for the units digit. Recall that a number is even if the units digit is 0,2,4,6 or 8.
Once we have the units digit selected, we have 6-1 = 5 choices for the first slot, 6-2 = 4 choices for the second slot, and so on until we get down to 6-5 = 1 choice for the fifth slot
We could write it out like this
- slot one = 5 choices
- slot two = 4 choices
- slot three = 3 choices
- slot four = 2 choices
- slot five = 1 choice
- slot six = units digit = 3 choices
Multiply those values out: 5*4*3*2*1*3 = 360
There are 360 different even numbers possible.
Answer:
Straight?
Step-by-step explanation:
Answer:
Multiply the cone's volume by 3
Step-by-step explanation:
Cylinder's Volume is
V = pi r ^2(h)
Cone's Volume is
V = 1/3 pi r^2 (h)
I'm guessing the series is supposed to be
![\displaystyle\sum_{n=1}^\infty\frac{n^2x^n}{7\cdot14\cdot21\cdot\cdots\cdot(7n)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Csum_%7Bn%3D1%7D%5E%5Cinfty%5Cfrac%7Bn%5E2x%5En%7D%7B7%5Ccdot14%5Ccdot21%5Ccdot%5Ccdots%5Ccdot%287n%29%7D)
By the ratio test, the series converges if the following limit is less than 1.
![\displaystyle\lim_{n\to\infty}\left|\frac{\frac{(n+1)^2x^{n+1}}{7\cdot14\cdot21\cdot\cdots\cdot(7n)\cdot(7(n+1))}}{\frac{n^2x^n}{7\cdot14\cdot21\cdot\cdots\cdot(7n)}}\right|](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cleft%7C%5Cfrac%7B%5Cfrac%7B%28n%2B1%29%5E2x%5E%7Bn%2B1%7D%7D%7B7%5Ccdot14%5Ccdot21%5Ccdot%5Ccdots%5Ccdot%287n%29%5Ccdot%287%28n%2B1%29%29%7D%7D%7B%5Cfrac%7Bn%5E2x%5En%7D%7B7%5Ccdot14%5Ccdot21%5Ccdot%5Ccdots%5Ccdot%287n%29%7D%7D%5Cright%7C)
The first
![n](https://tex.z-dn.net/?f=n)
terms in the numerator's denominator cancel with the denominator's denominator:
![\displaystyle\lim_{n\to\infty}\left|\frac{\frac{(n+1)^2x^{n+1}}{7(n+1)}}{n^2x^n}\right|](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cleft%7C%5Cfrac%7B%5Cfrac%7B%28n%2B1%29%5E2x%5E%7Bn%2B1%7D%7D%7B7%28n%2B1%29%7D%7D%7Bn%5E2x%5En%7D%5Cright%7C)
![|x^n|](https://tex.z-dn.net/?f=%7Cx%5En%7C)
also cancels out and the remaining factor of
![|x|](https://tex.z-dn.net/?f=%7Cx%7C)
can be pulled out of the limit (as it doesn't depend on
![n](https://tex.z-dn.net/?f=n)
).
![\displaystyle|x|\lim_{n\to\infty}\left|\frac{\frac{(n+1)^2}{7(n+1)}}{n^2}\right|=|x|\lim_{n\to\infty}\frac{|n+1|}{7n^2}=0](https://tex.z-dn.net/?f=%5Cdisplaystyle%7Cx%7C%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cleft%7C%5Cfrac%7B%5Cfrac%7B%28n%2B1%29%5E2%7D%7B7%28n%2B1%29%7D%7D%7Bn%5E2%7D%5Cright%7C%3D%7Cx%7C%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cfrac%7B%7Cn%2B1%7C%7D%7B7n%5E2%7D%3D0)
which means the series converges everywhere (independently of
![x](https://tex.z-dn.net/?f=x)
), and so the radius of convergence is infinite.
The answer is 93 years old.