# 1.3 + 3.5 + 5.7 +… + (2n –1) (2n + 1) = P(1) is true
Let P (k) be true 1.3 + 3.5 + 5.7 +… + (2k –1) (2k + 1) = To prove p(k+1) is true using P(k) lies
1.3 + 3.5 + 5.7 +… + (2k –1) (2k + 1)+[2(k+1)-1] [2(k+1)+1]
= = = = ……….(1)
= =  L. H.S of (1)
1.3 + 3.5 + 5.7 +… + (2k –1) (2k + 1)+[2(k+1)-1] [2(k+1)+1]
= = = = = Which is the R. H.s of (1) P(K+1) is true. By the Principle of mathematical induction, P(n) is true for all values of n where n N
Hence proved.

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