Use mathematical induction to prove the following statements are true

In summary, the conversation discusses using mathematical induction to prove a statement true for n ≥ 1. The statement is an equation and the attempt at a solution includes errors such as using n instead of k and not properly squaring the last term. It is suggested to use k and k+1 in the induction step and not mix it with n.
  • #1
Daaniyaal
64
0

Homework Statement


Use mathematical induction to prove the following statements are true for n≥1

a) 1^2+3^2+5^2+...+(2n-1)^2= [n(4n-1)]/3

Homework Equations





The Attempt at a Solution


Attempt at showing for n+1 is true:

n[4(n+1-1)]/3+ 2[k+1-1)^2
 
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  • #2
First off start by letting n=1 and check it. Then you can go to n+1.
 
  • #3
Daaniyaal said:

Homework Statement


Use mathematical induction to prove the following statements are true for n≥1

a) 1^2+3^2+5^2+...+(2n-1)^2= [n(4n-1)]/3

Homework Equations





The Attempt at a Solution


Attempt at showing for n+1 is true:

n[4(n+1-1)]/3+ 2[k+1-1)^2
First, what you are trying to prove true is an equation, and you have not "=". It looks to me like this is the right side of n= k with the new term added- but there are several errors.
First, bcause the term on the left is to be the "old" sum, you should not have "n+1" in place of n in 4(n-1)/3 but you should have 4(k-1)/3, not n. Second, in the sum the last term is (2n-1)^2- that is all squared so replacing n with k+ 1 gives (2(k+1)+1)^2 NOT "2[k+1-1]^2".

Since you want to prove the original statement is true for all n, it is a good idea to use "k" and "k+1" for the induction step. But do not mix "k" and "n".
 
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1. How does mathematical induction work?

Mathematical induction is a proof technique used to prove statements that are true for all natural numbers or positive integers. It involves two steps: the base case and the inductive step. The base case shows that the statement is true for the first natural number (usually 1), while the inductive step shows that if the statement is true for any natural number, then it must also be true for the next natural number. This process is repeated until the statement is proven to be true for all natural numbers.

2. When can mathematical induction be used?

Mathematical induction can be used to prove statements that are true for all natural numbers or positive integers. It can also be used to prove statements about other mathematical structures such as graphs, trees, and sets, as long as they have a well-defined ordering.

3. What is the difference between strong and weak induction?

In weak induction, the inductive step only assumes that the statement is true for the previous natural number. In strong induction, the inductive step assumes that the statement is true for all natural numbers up to the previous one. Strong induction is a more powerful version of weak induction and can be used to prove more complex statements.

4. Can mathematical induction be used to prove statements that are not true for all natural numbers?

No, mathematical induction can only be used to prove statements that are true for all natural numbers or positive integers. If a statement is not true for all natural numbers, then it cannot be proven using mathematical induction.

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

One common mistake is to assume that the statement is true for infinitely many natural numbers without showing the inductive step. Another mistake is to skip the base case or use the wrong base case. It is also important to make sure that the inductive step is valid and logically follows from the assumption that the statement is true for the previous natural number.

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