Question

Using principle of mathematical induction, prove that for all n ∈ N, 1/1.3+1/2.4+1/3.5+...+1/n(n+2)=1/2[3/2-(2n+3)/(n+1)(n+2)]


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Answer 1

Suppose P (n): 1.3 + 2.4 + 3.5 + … + n. (n + 2) = 1/6 n(n + 1) (2n + 7) Now let us check for n = 1, P (1): 1.3 = 1/6 × 1 × 2 × 9 : 3 = 3 P (n) is true for n = 1. Then, let us check for P (n) is true for n = k, and have to prove that P (k + 1) is true. P (k): 1.3 + 2.4 + 3.5 + … + k. (k + 2) = 1/6 k(k + 1) (2k + 7) … (i) Therefore, 1.3 + 2.4 + 3.5 + … + k. (k + 2) + (k + 1) (k + 3) Then, substituting the value of P (k) we get, = 1/6 k (k + 1) (2k + 7) + (k + 1) (k + 3) by using equation (i) = (k + 1) [{k(2k + 7)/6} + {(k + 3)/1}] = (k + 1) [(2k2 + 7k + 6k + 18)]/6 = (k + 1) [2k2 + 13k + 18]/6 = (k + 1) [2k2 + 9k + 4k + 18]/6 = (k + 1) [2k(k + 2) + 9(k + 2)]/6 = (k + 1) [(2k + 9) (k + 2)]/6 = 1/6 (k + 1) (k + 2) (2k + 9) P (n) is true for n = k + 1 Thus, P (n) is true for all n ∈ NRead more on Sarthaks.com - https://www.sarthaks.com/659371/prove-by-the-principle-of-mathematical-induction-1-3-2-4-3-5-n-n-2-1-6-n-n-1-2n-7

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