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.