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lesya692 [45]
2 years ago
8

Consider a group of kk people. Assume that each person's birthday is drawn uniformly at random from the 365 possibilities. (And

ignore leap years.) What is the smallest value of kk such that the expected number of pairs of distinct people with the same birthday is at least one?
Mathematics
1 answer:
QveST [7]2 years ago
6 0

Answer:

366

Step-by-step explanation:

Since there are 365 possible slots for people to have their birthday on, the worst case happens when all 365 people have different birthday. This means the 366th person would have their birthday falls on any of other’s birthday. Hence, kk must be at least 366.

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Assuming the series is

\displaystyle\sum_{n\ge1}\frac{x^n}{2n-1}

The series will converge if

\displaystyle\lim_{n\to\infty}\left|\frac{\frac{x^{n+1}}{2(n+1)-1}}{\frac{x^n}{2n-1}}\right|

We have

\displaystyle\lim_{n\to\infty}\left|\frac{\frac{x^{n+1}}{2(n+1)-1}}{\frac{x^n}{2n-1}}\right|=|x|\lim_{n\to\infty}\frac{\frac1{2n+1}}{\frac1{2n-1}}=|x|-\lim_{n\to\infty}\frac{2n-1}{2n+1}=|x|

So the series will certainly converge if -1, but we also need to check the endpoints of the interval.

If x=1, then the series is a scaled harmonic series, which we know diverges.

On the other hand, if x=-1, by the alternating series test we can show that the series converges, since

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Please help me with these two math problems that I do not quite understand. This is urgent!
g100num [7]

Answer:

1)a_{n}=a_{1}+(n-1)d\Rightarrow a_{n}=-1-2(n-1)\\a_{2}=-1-2(2-1)\Rightarrow a_{2}=-3\\a_{3}=-1-2(3-1)\Rightarrow a_{3}=-5\\(...)\\a_{10}=-1-2(10-1)=-19

2) a_{10}=4+8(10-1)\Rightarrow a_{10}=76

Step-by-step explanation:

1) To write an Arithmetic Sequence, as an Explicit Term, is to write a general formula to find any term for this sequence following this pattern:

a_{n}=a_{1}+d(n-1)\Rightarrow \left\{\begin{matrix}a_{n}=n^{th}\: term\\a_1=1st \: term \\d=\: difference\\n=n^{th}\, term\end{matrix}\right.

<em>"Write an explicit formula for each explicit formula A(n)=-1+(n-1)(-2)"</em>

This isn't quite clear. So, assuming you meant

Write an explicit formula for each term of this sequence A(n)=-1+(n-1)(-2)

As this A(n)=-1+(n-1)(-2)  is already an Explicit Formula, since it is given the first term a_{1}=-1 the common difference d=-2 let's find some terms of this Sequence through this Explicit Formula:

a_{n}=a_{1}+(n-1)d\Rightarrow a_{n}=-1-2(n-1)\\a_{2}=-1-2(2-1)\Rightarrow a_{2}=-3\\a_{3}=-1-2(3-1)\Rightarrow a_{3}=-5\\(...)\\a_{10}=-1-2(10-1)=-19

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Then, just plug in the first term and the common difference into the explicit formula:

a_{10}=4+8(10-1)\Rightarrow a_{10}=76

6 0
2 years ago
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