Binomial Theorem proof by induction, Spivak

In summary: This gives: \displaystyle \sum_{j=1}^{n+1} \binom{n}{j} = \binom{n+1}{j-1}\,a^{(n+1-j)}\,b^{\,j} \,.aspx?Here, we have used the binomial theorem to solve for the sum of two binomials. The first binomial is \displaystyle \binom{n}{j-1} + \binom{n}{j}, and the second binomial is \displaystyle \binom{n}{j+1} + \binom{n}{j-1}. First, we use the distributive law to combine
  • #1
usn7564
63
0
Homework Statement
Prove the binomial theorem by induction.

The attempt at a solution
http://desmond.imageshack.us/Himg35/scaled.php?server=35&filename=sumu.png&res=landing [Broken]

Hi, running into trouble with this proof and google hasn't helped me. I don't understand the jump here, and as it's not really explained in the videos/.pdf's I found I presume it's something really simple that I'm just not getting.

I saw one explanation saying "just put k=j+1, put k in the equation in place of j then switch back to j" and while it worked I'm not seeing why I can replace j with k, as k=/= j.

Any input would be appreciated, this is really puzzling me.
 
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  • #2
usn7564 said:
Homework Statement
Prove the binomial theorem by induction.

The attempt at a solution
http://desmond.imageshack.us/Himg35/scaled.php?server=35&filename=sumu.png&res=landing [Broken]

Hi, running into trouble with this proof and google hasn't helped me. I don't understand the jump here, and as it's not really explained in the videos/.pdf's I found I presume it's something really simple that I'm just not getting.

I saw one explanation saying "just put k=j+1, put k in the equation in place of j then switch back to j" and while it worked I'm not seeing why I can replace j with k, as k=/= j.

Any input would be appreciated, this is really puzzling me.

j and k are just dummy variables in the sum. Once you've replaced j with k, and worked out the form of the summand and the bounds, you can replace k with anything you like, including j. It's OK as long as you replace every instance of it.

Try replacing the "k"s with the actual numbers and write out the sum. Do the same with the "j"s and you'll find out the come out to be the sum of the same terms, proving they're equal.
 
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  • #3
Ah, right. Thought I wouldn't be able to without doing it on LHS as well, suppose it shouldn't make a difference though. I'll try putting some numbers in and hopefully have it 100% clear. Thanks!
 
  • #4
I think you will need to use the following identity:
[tex]\binom{n}{j-1} + \binom{n}{j} = \binom{n+1}{j}[/tex]

It shouldn't be hard to prove if you haven't seen it before.
 
  • #6
usn7564 said:
...

http://desmond.imageshack.us/Himg35/scaled.php?server=35&filename=sumu.png&res=landing [Broken]
...
This may not be anymore helpful to you than the above posts, but I'll add my two cents.

Your question boils down to:
How is [itex]\displaystyle \sum_{j=0}^{n} \binom{n}{j}\,a^{(n-j)}\,b^{\,(j+1)} \ [/itex] equal to [itex]\displaystyle \sum_{j=1}^{n+1} \binom{n}{j-1}\,a^{(n+1-j)}\,b^{\,j} \ ?[/itex]​

Starting with [itex]\displaystyle \sum_{j=0}^{n} \binom{n}{j}\,a^{(n-j)}\,b^{\,(j+1)} \,,[/itex] let k = j+1. Then j = k-1 . The sum starts with j = 0, which corresponds to k = 1 . The sum terminates with j = n, which corresponds to k = n+1 .
Replacing j with k-1 gives: [itex]\displaystyle \sum_{k=1}^{n+1} \binom{n}{k-1}\,a^{(n-(k-1))}\,b^{\,(k-1+1)} \,.[/itex]

Simplifying this, we have: [itex]\displaystyle \sum_{k=1}^{n+1} \binom{n}{k-1}\,a^{(n+1-k)}\,b^{\,k} \,.[/itex]

Now, since k is a "dummy" variable, replace it with j.
 
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1. What is the Binomial Theorem?

The Binomial Theorem is a mathematical formula that allows us to expand a binomial expression, such as (a + b)^n, into a polynomial with n+1 terms. It is often used in algebra, calculus, and probability to simplify and solve equations involving binomial expressions.

2. What is the proof by induction for the Binomial Theorem?

The proof by induction for the Binomial Theorem is a mathematical proof technique used to demonstrate that the formula holds true for all possible values of n. It involves proving the formula for the base case (usually n = 0 or n = 1) and then assuming it is true for some arbitrary value of n. Using this assumption, we can then prove that the formula holds true for n+1. This process is repeated until we can prove that the formula holds true for all values of n.

3. Who is Spivak and what is his contribution to the proof by induction for the Binomial Theorem?

Michael Spivak is a mathematician and textbook author who is best known for his book "Calculus", which is widely used in universities and colleges. In this book, he presents a rigorous and detailed proof by induction for the Binomial Theorem, which has become a standard reference for students and mathematicians alike.

4. Why is proof by induction important for the Binomial Theorem?

Proof by induction is important for the Binomial Theorem because it allows us to confidently apply the formula to any value of n without having to manually expand each expression. It also helps us understand the underlying principles and patterns of the formula, making it easier to use in more complex equations.

5. What are some real-life applications of the Binomial Theorem?

The Binomial Theorem has many applications in various fields, such as finance, physics, and statistics. For example, it can be used to calculate the probability of certain events in a given number of trials, or to simplify complicated equations in physics involving binomial expressions. In finance, it is used to calculate compound interest and binomial option pricing.

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