Prove the following by the principle of mathematical induction:
1.3 + 3.5 + 5.7 + … + (2n – 1) (2n + 1)
Let P(n): 1.3 + 3.5 + 5.7 + … + (2n – 1) (2n + 1)
For n = 1
P(1): (2.1 – 1) (2.1 + 1) =
= 1x3 =
= 3 = 3
Since, P(n) is true for n =1
Now, For n = k, So
1.3 + 3.5 + 5.7 + … + (2k – 1) (2k + 1) - - - - - - - (1)
We have to show that,
1.3 + 3.5 + 5.7 + … + (2k – 1) (2k + 1) + (2k + 1)(2k + 3)
Now,
1.3 + 3.5 + 5.7 + … + (2k – 1) (2k + 1) + (2k + 1)(2k + 3)
= + (2k + 1)(2k + 3) using equation (1)
=
=
=
=
=
=
Therefore, P(n) is true for n=k + 1
Hence, P(n) is true for all n∈ N by PMI