Prove by induction that 1+3+5+7+⋯+(2n-1)=n^2

Given

1 + 3 + 5 + 7 + ⋯ + (2n-1) = n2 ∀ n ∈ N

Proof

We will prove the statement using mathematical induction.

STEP 1

In this step, we check if the statement is true for n = 1.

1 + 3 + 5 + 7 + ⋯ + (2n-1) = 1

n2 = (1)2 = 1

LHS = RHS

The statement is true for n = 1.

STEP 1

Let the statement be true for n = k ∀ k ∈ N.

1 + 3 + 5 + 7 + ⋯ + ( 2k – 1 ) = k2

Now, we need to check if the statement is true for n = k + 1 given that it is true for n = k.

Basically we need to prove,

1 + 3 + 5 + 7 + ⋯ + ( 2( k + 1 ) – 1 ) = ( k + 1 )2

We start from LHS

LHS = 1 + 3 + 5 + 7 + ⋯ + ( 2( k + 1 ) – 1 )

= 1 + 3 + 5 + 7 + ⋯ + ( 2k – 1 ) + ( 2( k + 1 ) – 1 )

= ( 1 + 3 + 5 + 7 + ⋯ + ( 2k – 1 ) ) + ( 2( k + 1 ) – 1 )

Substituting k2 in the above equation

= k2 + ( 2( k + 1 ) – 1 )

= k2 + 2k + 1

= ( k + 1 )2

= RHS

Therefore, we can say that the given statement is true for n = k + 1 whenever it is true for n = k ∀ k ∈ N. By principle of mathematical induction the given statement is true for ∀ n ∈ N.

Hence Proved.

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