Mathematical Induction - Questions and Solutions

Mathematical Induction
Serial order wise

Transcript

Question 4: Prove the following by using the principle of mathematical induction for all n â N: 1.2.3 + 2.3.4 + âĶ + n(n + 1) (n + 2) = (ð(ð + 1)(ð + 2)(ð + 3))/4 Let P(n): 1.2.3 + 2.3.4 + âĶ + n(n + 1) (n + 2) = (ð(ð + 1)(ð + 2)(ð + 3))/4 For n = 1, L.H.S = 1.2.3 = 6 R.H.S = (1(1+1)(1+2)(1+3))/4 = (1 Ã 2 Ã 3 Ã 4)/4 = 6 L.H.S. = R.H.S âīP(n) is true for n = 1 Assume P(k) is true 1.2.3 + 2.3.4 + âĶ + k(k + 1) (k + 2) = (ð(ð + 1)(ð + 2)(ð + 3))/4 We will prove that P(k + 1) is true. 1.2.3 + 2.3.4 + âĶ + (k + 1) ((k +1) + 1) ((k + 1) + 2) = (ð + 1)((ð + 1)+ 1)((ð + 1)+ 2)((ð + 1)+ 3)/4 1.2.3 + 2.3.4 + âĶ + (k + 1) (k + 2) (k + 3) = ((ð + 1)(ð + 2)(ð + 3)(ð + 4))/4 1.2.3 + 2.3.4 + âĶ + k(k + 1) (k + 2) +(k + 1) (k + 2) (k + 3) = ((ð + 1)(ð + 2)(ð + 3)(ð + 4))/4 We have to prove P(k+1) from P(k) i.e. (2) from (1) From (1) 1.2.3 + 2.3.4 + âĶ + k(k + 1) (k + 2) = (ð(ð + 1)(ð + 2)(ð + 3))/4 Adding (k+1) (k+2) (k+3) both sides 1.2.3 + 2.3.4 + âĶ + k(k + 1) (k + 2) +(k + 1) (k + 2) (k + 3) = (ð(ð + 1)(ð + 2)(ð + 3))/4 +(k + 1) (k + 2) (k + 3) = (ð(ð + 1)(ð + 2)(ð + 3) + 4(ð + 1)(ð + 2)(ð + 3))/4 = ((ð + 1)(ð + 2)(ð + 3) (ð + 4))/4 Thus, 1.2.3 + 2.3.4 + âĶ + k(k + 1) (k + 2) +(k + 1) (k + 2) (k + 3) = ((ð + 1)(ð + 2)(ð + 3)(ð + 4))/4 which is the same as P(k + 1) âī P(k + 1) is true whenever P(k) is true. âī By the principle of mathematical induction, P(n) is true for n, where n is a natural number