Question -
Answer -
Let P (n): 1 + 3 + 5 + … + (2n – 1) = n2
Let us check P (n) is true for n = 1
P (1) = 1 =12
1 = 1
P (n) is true for n = 1
Now, Let’s check P (n) is true for n = k
P (k) = 1 + 3 + 5 + … + (2k – 1) = k2 … (i)
We have to show that
1 + 3 + 5 + … + (2k – 1) + 2(k + 1) – 1 = (k + 1)2
Now,
1 + 3 + 5 + … + (2k – 1) + 2(k + 1) – 1
= k2 + (2k + 1)
= k2 + 2k + 1
= (k + 1)2
P (n) is true for n = k + 1
Hence, P (n) is true for all n ∈ N.