MHB Solve the Binomial Theorem Puzzle: Find Missing Member

Alexstrasuz1
Messages
20
Reaction score
0
Screenshot by Lightshot
The translation in binom coefficent of 4th and 10th are mathching each other.
Find the member which doesn't have x in it.
I understand all of it but the part where (n up n-3)=(n up 9) I just don't understand how they got 12 here
 
Mathematics news on Phys.org
Alexstrasuz said:
Screenshot by Lightshot
The translation in binom coefficent of 4th and 10th are mathching each other.
Find the member which doesn't have x in it.
I understand all of it but the part where (n up n-3)=(n up 9) I just don't understand how they got 12 here
I'm not sure what the first question is asking.

For the second we would have to have
[math](x^2)^i \cdot \left ( \frac{1}{x} \right ) ^{n - i} = 1[/math]

Which leads to 2i = n - i thus i = n/3. So this will only happen for n divisible by 3.

For the third you have
[math]{n \choose n - 3} = {n \choose 9}[/math]

We can simply compare the bottom element on both sides and conclude that n - 3 = 9. On the other hand if we use the definition of the binomial coefficient we get the equation
[math](n - 3) \cdot (n - 4) \cdot (n - 5) \cdot (n - 6) \cdot (n - 7) \cdot (n - 8) = 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4[/math]

Again by comparison we can find that n = 12. On the other hand this is a 6th degree polynomial in n and can potentially have at least two real solutions since we know it already has one. I don't know how to analyze this in general, but WA gives another real root as n = -1. The other four (WA missed one) are complex.

-Dan
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.

Similar threads

Replies
2
Views
2K
Replies
5
Views
1K
Replies
5
Views
2K
Replies
30
Views
4K
Replies
9
Views
2K
Replies
4
Views
6K
Replies
3
Views
4K
Replies
2
Views
2K
Back
Top