Q. 16
Prove the following by the principle of mathematical induction:
12 + 32 + 52 + … + (2n – 1)2
Answer :
Let P(n): 12 + 32 + 52 + … + (2n – 1)2 = 
For n = 1
= (2.1 – 1)2 = 
= 1 = 1
Since, P(n) is true for n = 1
Let P(n) is true for n = k ,
P(k) ): 12 + 32 + 52 + … + (2k – 1)2 = 
- - (1)
We have to show that,
12 + 32 + 52 + … + (2k – 1)2 + (2k + 1)2 = 
Now,
{12 + 32 + 52 + … + (2k – 1)2} + (2k + 1)2
= 
using equation (1)
= 
= (2k + 1)
= (2k + 1)
= (2k + 1)
= 
= 
= 
= 
= 
= 
Therefore, P(n)is true for n = k + 1
Hence, P(n) is true for all n∈ N
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