Question

1+3+5+....+(2n-1)=n^2​

Answer 1

Proof by induction on n:

 

Step 1:  prove that the equation is valid when n = 1

 

              When n = 1, we have (2(1) - 1) = 12, so  the statement                        holds for n = 1.

 

Step 2:  Assume that the equation is true for n, and prove that the                   equation is true for n + 1.

 

              Assume:  1 + 3 + 5 + ... + (2n - 1) = n2

 

             Prove:  1 + 3 + 5 +...+ (2(n + 1) - 1) = (n + 1)2

 

                 Proof:  1 + 3 + 5 +... + (2(n + 1) - 1) 

                              = 1 + 3 + 5 + ... + (2n - 1) + (2n + 2 - 1)

                              = n2 + (2n + 2 - 1)  (by assumption)

                              = n2 + 2n + 1

                              = (n + 1)2

 

So, by induction, for every positive integer n,

1 + 3 + 5 + ... + (2n - 1) = n2

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