By using the principle of mathematical induction prove that 1.2.3+ 2.3.4+ 3.4.5 .... +n(n+ 1)(n+ 2) = n(n +1)(n+ 2)(n+ 3) /4

By using the principle of mathematical induction prove that 1.2.3+ 2.3.4+ 3.4.5 .... +n(n+ 1)(n+ 2) = n(n +1)(n+ 2)(n+ 3) /4

Solution

Let P(n) : 1.2.3 + 2.3.4 + 3.4.5 ... + n(n + 1)(n + 2) = n(n + 1)(n + 2)(n + 3) / 4

For n = 1
L. H. S. = 6 and R. H. S. = 6.
Therefore, L. H. S. = R. H. S. i.e., P(1) is true.
Let us suppose P(k) is true.
Therefore, putting n = k, we have,
1.2.3 + 2.3.4 + 3.4.5 .... +k(k + 1)(k + 2) = k(k + 1)(k + 2)(k + 3) / 4. ................................................(1)
Adding (k + 1)(k + 2)(k+ 3) to both sides of (1), we have,
1.2.3 + 2.3.4 + 3.4.5 ... + k(k + 1)(k + 2) + (k + 1)(k + 2)(k + 3) = {k(k + 1)(k + 2)(k + 3) /4}  + (k 1)(k 2)(k 3).
                                                                                                     =(k + 1)(k + 2)(k + 3) {(k/4) + 1}.
                                                                                                     =(k + 1)(k + 2)(k + 3)(k + 4) / 4.
                                                                                                    =(k+ 1)(k+ 1+ 1)(k +1 +2)(k+ 1+ 3) / 4.
Here P(n) is true for n = k+ 1 i.e. P(k+ 1) is true.
Therefore, By the principle of mathematical induction, P(n) is true for all natural numbers n.