- #1
jdinatale
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I immediately thought of induction, so that is what I used, but I can't seem to make any progress past a certain point.
Trying to prove this equality involving a summation of a binomial coefficient.
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In summary, The conversation is about a person seeking help with an induction proof. They tried multiplying the LHS by n+1 but couldn't progress further. Another person suggests putting m = k+1 in the third line, which leads to solving the problem. The reason for this is that the RHS of the equation is 2n+1, which can be expressed as ∑ n+1Cr.
- #2
tiny-tim
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hi jdinatale! 
(i haven't looked at your induction proof , but …)
why not just multiply the LHS by n+1 ?
(i haven't looked at your induction proof , but …)
why not just multiply the LHS by n+1 ?
- #3
jdinatale
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tiny-tim said:hi jdinatale!
(i haven't looked at your induction proof , but …)
why not just multiply the LHS by n+1 ?
Ok, I tried that and I eventually could not go any further. Any ideas on what's going wrong?
- #4
tiny-tim
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hi jdinatale!
in the third line you have ∑k=0…j j+1Ck+1
put m = k+1, that's ∑m=1…j+1 j+1Cm …
in the third line you have ∑k=0…j j+1Ck+1
put m = k+1, that's ∑m=1…j+1 j+1Cm …
what is that?
- #5
jdinatale
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tiny-tim said:hi jdinatale!
in the third line you have ∑k=0…j j+1Ck+1
put m = k+1, that's ∑m=1…j+1 j+1Cm …
what is that?
Brilliant! I've solved the problem now, thank you so much. But please tell me, how did you possibly know to do that? That was not obvious to me at all, and I'm not sure how you would just know to do that.
- #6
tiny-tim
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easy!
the clue was in the question …
the clue was in the question …
the RHS said 2n+1,
which i know is ∑ n+1Cr
which i know is ∑ n+1Cr
1. What is a binomial coefficient?
A binomial coefficient is a mathematical term that represents the number of ways to choose a subset of size k from a larger set of size n. It is denoted by n choose k and is calculated by the formula n choose k = n! / (k! * (n-k)!).
2. What does it mean to prove an equality involving a summation of a binomial coefficient?
Proving an equality involving a summation of a binomial coefficient means to show that two expressions are equal by using algebraic manipulations and mathematical identities. It requires showing that both sides of the equation simplify to the same value.
3. How do you approach proving an equality involving a summation of a binomial coefficient?
The key to proving an equality involving a summation of a binomial coefficient is to use known mathematical identities and properties, such as the binomial theorem and properties of factorials. It is also important to carefully manipulate the expressions on both sides of the equation to show that they are equivalent.
4. Can you provide an example of an equality involving a summation of a binomial coefficient?
One example of an equality involving a summation of a binomial coefficient is the Vandermonde's identity, which states that for any non-negative integers m and n, the following equation holds true: n + m choose k = Σj=0k (n choose j) * (m choose k-j).
5. What are some real-world applications of proving equalities involving summations of binomial coefficients?
Equalities involving summations of binomial coefficients are often used in probability and statistics, as well as in various areas of mathematics such as combinatorics and number theory. They can also be applied in computer science, particularly in algorithms and data structures. For example, the binomial coefficient can be used to calculate the number of possible combinations in a given set, which is useful in generating unique passwords or in analyzing the complexity of algorithms.
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