Answer:
See proof below
Step-by-step explanation:
The inductive proof consists on the following steps:
1) Base case: for n=0, we will prove that Γ(1)=0!=1. We have that
![\gamma(1)=\int_{0}^{\infty}t^{1-1}e^{-t}dt=\int_{0}^{\infty}t^{1-1}e^{-t}dt](https://tex.z-dn.net/?f=%5Cgamma%281%29%3D%5Cint_%7B0%7D%5E%7B%5Cinfty%7Dt%5E%7B1-1%7De%5E%7B-t%7Ddt%3D%5Cint_%7B0%7D%5E%7B%5Cinfty%7Dt%5E%7B1-1%7De%5E%7B-t%7Ddt)
![=-e^{-t}|^{\infty}_{0}=0+e^{0}=1](https://tex.z-dn.net/?f=%3D-e%5E%7B-t%7D%7C%5E%7B%5Cinfty%7D_%7B0%7D%3D0%2Be%5E%7B0%7D%3D1)
Hence the base case holds.
2) Inductive step: suppose that Γ(n + 1) = n! for some natural number n. We will prove that Γ((n + 1)+1) = (n+1)!
Use integration by parts, with the following parts:
u=t^{n+1}, du=(n+1)t^n
dv=e^{-t}, v=-e^{-t}
![=(n+1)\gamma(n+1)=(n+1)n!=(n+1)!](https://tex.z-dn.net/?f=%3D%28n%2B1%29%5Cgamma%28n%2B1%29%3D%28n%2B1%29n%21%3D%28n%2B1%29%21)
and we used the induction hypotheses on this last line. Also, -t^n e^-t tends to zero as n tends to infiity (the exponential decays faster than any polynomial).
We have proved the statement for n+1, and by mathematical induction, the statement holds for all n.
The number of bacteria in a culture is increasing according to the law of exponential growth. There are 125 bacteria in the culture after 2 hours and 350 bacteria after 4 hours.
Answer:
nice
Step-by-step explanation:
Answer:
I believe it is A
Step-by-step explanation:
because if the time is 24.78 and you have to get the exact time then then answer is A but if it asks you to round to the nearest millisecond then then its B. Sorry if its confusing hope this helps! :)
1 light year = 5.879 x 10^12
So approximately, the answer is C