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 the principle of mathematical induction the given statement is true for ∀ n ∈ N.
Hence Proved.