Prove that 1 + 3 + 5 + … + (2n – 1) = n2 using the principle of Mathematical induction.
Answers
Need-to-know
- Mathematical Induction(Proofs)
It is used to show a claim. It is used when proof over natural numbers is enough.
The proof is not perfect, though. It is likened to the dominos.
Proof
Claim:
Let's check if works.
Now, let's see if works, the reason is is the next integer of . Only one value that works is enough, because
This proves every next integer of works. Hence, it is proven for all the natural numbers.
More information
- Series
is the series.
- Facts
The mathematical induction is not actually inductive. It is deductive because the conclusion comes from normal facts.
If it were inductive, we would have proven for every natural number.
Given :
To Find :
- Prove the given equation using Principal of mathematical induction
Method :
Let be a given statement for
Initial Step : Let the statement is true for is true and
Inductive Step : If the statement is true for (where k is a particular but arbitrary natural number) then the statement is true for truth of implies the truth of . Then
Solution :
Let the given statement be defined as
Then is true for all natural numbers
Step 1 :-
Step 2 :- Let us assume that the statement is true for
Step 3 :- To prove that is true.
is true whenever is true.
is true for all