Inductive proof of summation formula

In summary, the homework statement is trying to prove the summation formula using induction. The first inductive step is to prove that the sum of fractions with the same denominator is equal to 1. The second inductive step is to prove that the sum of fractions with the same denominator is equal to 1 - 1/n+1.
  • #1
livcon
6
0

Homework Statement


Prove by induction the following summation formula:
[tex]\frac{1}{1\1*2} + \frac{1}{2*3} + ... + \frac{1}{n(n+1)} = 1 - \frac{1}{n+1}[/tex]
[tex]n \geq 1[/tex]

Homework Equations


-

The Attempt at a Solution


Inductive step:
1. [tex]\frac{1}{1*2} + \frac{1}{2*3} + ... + \frac{1}{n(n+1)} + \frac{1}{(n+1)((n+1)+1)} = 1 - \frac{1}{(n+1)+1}[/tex]

2. [tex]\frac{1}{1*2} + \frac{1}{2*3} + ... + \frac{1}{n(n+1)} + \frac{1}{(n+1)((n+1)+1)} = 1 - \frac{1}{n+1} + \frac{1}{(n+1)(n+2)}[/tex]

To prove this (1) should equal (2) (right?), but I don't see how to manipulate (2) to make it equal to (1)

Thanks in advance!
 
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  • #2
Common denominator!
 
  • #3
but I don't see how to manipulate (2) to make it equal to (1)
You don't need to. All you need to do is to show these two sums of fractions are equal.
 
  • #4
That's right, but in this case I've actually been staring at the problem for a while, so I'm really interested in how to do the algebra in (2). Thanks
 
  • #5
As a warmup exercise, prove the following:

[tex]\frac{1}{3} + \frac{1}{6} = \frac{3}{2} - 1[/tex]
 
  • #6
livcon said:
That's right, but in this case I've actually been staring at the problem for a while, so I'm really interested in how to do the algebra in (2). Thanks

see how 1 - 1/n+1 looks first
 
  • #7
I appreciate the help, but I'm afraid I'm still blind here, proving the equivalence in the warmup exercise is easy Hurkyl, but I just don't see it in my problem.
 
  • #8
livcon said:
I appreciate the help, but I'm afraid I'm still blind here, proving the equivalence in the warmup exercise is easy Hurkyl, but I just don't see it in my problem.
Isn't proving
[tex]1 - \frac{1}{(n+1)+1} = 1 - \frac{1}{n+1} + \frac{1}{(n+1)(n+2)}[/tex]​
exactly the same kind of problem, though? Why can't you do the same thing you did for the warmup problem?
 
  • #9
I guess it's because in the warmup exercise I could easily add and subtract the fractions because it was all constants, but in the one involving the variable n I got stuck. So I would probably have an epiphany if someone would just write it out. This is just a matter of me not seeing the obvious steps in this particular problem.
 
  • #10
What do you get when you change
[tex]1- \frac{1}{n+1}+ \frac{1}{(n+1)(n+2)}[/tex]
to fractions having the same denominator?

That's what everyone has been telling you do to. Have you done it?
 

1. What is an inductive proof of summation formula?

An inductive proof of summation formula is a mathematical method used to prove that a certain formula or equation holds true for all possible values within a given range. It involves using mathematical induction, which is a technique of proving a statement by showing that it holds true for the first value and then showing that if it holds true for one value, it also holds true for the next value.

2. How does an inductive proof of summation formula work?

An inductive proof of summation formula works by starting with the base case, which is the first value or the smallest value in the given range. This base case is then used to show that the statement holds true for the next value in the range. This process is repeated until the entire range is covered and the statement is proven to hold true for all values within the range.

3. What is the importance of an inductive proof of summation formula?

An inductive proof of summation formula is important because it provides a rigorous and systematic way to prove that a given formula or equation is true for all possible values within a range. This is particularly useful in mathematics and science, where accuracy and precision are crucial.

4. What are some examples of inductive proofs of summation formula?

Some examples of inductive proofs of summation formula include proving that the sum of the first n positive integers is n(n+1)/2, or that the sum of the first n odd numbers is n^2. These proofs involve using mathematical induction to show that the formula holds true for all possible values of n.

5. Are there any limitations to using inductive proof of summation formula?

Yes, there are limitations to using inductive proof of summation formula. This method can only be used to prove statements that hold true for discrete values, such as integers. It cannot be used for continuous values, such as real numbers. Additionally, this method can be time-consuming and may not always be the most efficient way to prove a statement.

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