Proving the Sum of Reciprocals by Induction

In summary, the conversation is about a homework problem involving proving a summation by induction. The problem involves the equation \frac{1}{1(2)} + 1\frac{1}{2(3)}+...+\frac{1}{n(n+1)} = \frac{n}{n+1}. The base case is n=1 and the attempt at a solution involves breaking down the summation into smaller parts and simplifying it. The conversation ends with the person asking for help in completing the proof.
  • #1
thefeedinghan
2
0
Hello there, I'm having trouble proving this by induction

Homework Statement


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


Homework Equations


For the base case n=1

[itex]\frac{1}{1(2)}=\frac{1}{1+1} = \frac{1}{2} = \frac{1}{2}[/itex]

[itex]\frac{k}{k+1} + \frac{1}{(k+1)(k+2)}[/itex] <- the second term would be the next integer

The Attempt at a Solution


[itex]\frac{1}{1(2)}+ \frac{1}{2(3)}+\frac{1}{k(k+1)} + \frac{1}{(k+1)(k+2)} = \frac{k+1}{(k+1)+1} = \frac{k}{k+2} + \frac{1}{k+2}[/itex]

I don't know where to go from here, any help would be appreciated
 
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  • #2
welcome to pf!

hi welcome to pf! :wink:
thefeedinghan said:
[itex]\frac{k}{k+1} + \frac{1}{(k+1)(k+2)}[/itex]

ok, now put the whole thing over (k+1)(k+2) …

what do you get? :smile:
 
  • #3
oh, thanks very much! was pretty obvious now!
 

1. What is induction proof?

Induction proof is a mathematical method used to prove that a certain statement is true for all natural numbers. It involves proving that the statement is true for the base case (usually n=1) and then showing that if it is true for any arbitrary natural number, it must also be true for the next natural number.

2. Why is induction proof important?

Induction proof is an important tool in mathematics, as it allows us to prove that a statement is true for an infinite number of cases, without having to test each one individually. It is also used in computer science and physics to prove the correctness of algorithms and theories.

3. What are the steps to perform an induction proof?

The steps for an induction proof are as follows:1. Prove the statement is true for the base case.2. Assume the statement is true for an arbitrary natural number.3. Use this assumption to prove that the statement is also true for the next natural number.4. Conclude that the statement is true for all natural numbers by the principle of mathematical induction.

4. What are some common mistakes to avoid in induction proof?

Some common mistakes to avoid in induction proof include:1. Not proving the statement for the base case.2. Assuming the statement is true for all natural numbers instead of just the next natural number.3. Using circular reasoning or assuming the statement is true without proper justification.4. Skipping steps or not providing enough detail in the proof.

5. Can induction proof be used to prove all statements?

No, induction proof can only be used to prove statements that are true for all natural numbers. It cannot be used for statements involving real numbers or other types of numbers. Additionally, some statements may require alternative methods of proof and may not be suitable for induction.

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