Mathematical Induction simplification

In summary, the conversation discusses a mathematical induction question and how to simplify the solution, which is (k+2)!-1. The suggestion is to rearrange and factor out (k+1)! in order to solve the equation. Further guidance is given to help reach the correct answer.
  • #1
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Homework Statement



I have been workin on a mathematical induction question and have run into trouble with the simplification.



Homework Equations





The Attempt at a Solution



I know that the solution i am trying to reach is (k+2)!-1
but i do not know where to go from the equation:
(k+1)!-1 + (k+1)(k+1)! could someone please help.
 
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  • #2
Chadlee88 said:
(k+1)!-1 + (k+1)(k+1)! could someone please help.

rearrange - put the -1 at the end and then look closely at the two terms now together.

If you require more help please show what you have tried.
 
  • #3
if in doubt always write out what the factorial really means
 
  • #4
Try to factor out (k + 1)!, and see if you can get the answer. Note that:
(n + 1) n! = (n + 1)!
Can you go from here? :)
 
  • #5
(k+1)!-1 + (k+1)(k+1)!= (k+1)!+(k+1)(k+1)!-1, so here as the
others explained you need to factor out (k+1)!, so you will get
(k+1)!(1+k+1)-1=(k+2)(k+1)!-1, so now look at what vietdao said, and you will come to the right answer.
 

What is mathematical induction simplification?

Mathematical induction simplification is a method used to prove mathematical statements or formulas that involve an infinite number of cases. It involves breaking down the problem into smaller, more manageable cases and using logical reasoning to prove that the statement holds for all cases.

Why is mathematical induction simplification important?

Mathematical induction simplification is important because it allows us to prove mathematical statements that would be difficult or impossible to prove otherwise. It is a powerful tool in the world of mathematics and is used to solve complex problems and prove theorems.

How does mathematical induction simplification work?

Mathematical induction simplification works by breaking down a problem into smaller cases and proving that the statement holds for each case. This is done by showing that if the statement holds for a particular case, it also holds for the next case until all cases have been exhausted.

What are the steps involved in mathematical induction simplification?

The steps involved in mathematical induction simplification are:

  • Step 1: Prove the statement holds for the first case (usually the base case).
  • Step 2: Assume the statement holds for a particular case.
  • Step 3: Use this assumption to prove that the statement holds for the next case.
  • Step 4: Repeat step 3 until all cases have been exhausted.
  • Step 5: Conclude that the statement holds for all cases.

What are some common mistakes to avoid when using mathematical induction simplification?

Some common mistakes to avoid when using mathematical induction simplification are:

  • Not proving the base case or assuming it to be true without proof.
  • Using incorrect or incomplete assumptions to prove the statement for the next case.
  • Assuming that the statement holds for all cases after only proving it for a few cases.
  • Using circular reasoning or making invalid logical jumps.

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