Prove the following by the principle of mathematical induction:
2 + 5 + 8 + 11 + … + (3n – 1) = 1/2 n(3n + 1)
Let P(n): 2 + 5 + 8 + 11 + … + (3n – 1) = n(3n + 1)
For n=1
P(1): 2 = .1.(4)
2 = 2
Since, P(n) is true for n = 1
Let P(n) is true for n = k, so
P(k): 2 + 5 + 8 + 11 + … + (3k – 1) = k(3k + 1) - - - - - - - (1)
We have to show that,
2 + 5 + 8 + 11 + … + (3k – 1) + (3k + 2) = (k + 1)(3k + 4)
Now,
{2 + 5 + 8 + 11 + … + (3k – 1)} + (3k + 2)
= .k(3k + 1) + (3k + 2)
=
=
=
=
=
=
Therefore, P(n) is true for n = k + 1
Hence, P(n) is true for all n ∈ N by PMI