Prove by the principle of mathematical induction:1/1.2 + 1/2.3 + 1/3.4

1.	Prove by the principle of mathematical induction:1/1.2 + 1/2.3 + 1/3.4 + … + 1/n(n + 1) = n/(n + 1)
Answer» Suppose P (n) = 1/1.2 + 1/2.3 + 1/3.4 + … + 1/n(n+1) = n/(n+1) For, n = 1 P (n) = 1/1.2 = 1/1+1 1/2 = 1/2 P (n) is true for n = 1 Now, let’s check for P (n) is true for n = k, 1/1.2 + 1/2.3 + 1/3.4 + … + 1/k(k+1) + k/(k+1) (k+2) = (k+1)/(k+2) Now, 1/1.2 + 1/2.3 + 1/3.4 + … + 1/k(k + 1) + k/(k + 1) (k + 2) = 1/(k + 1)/(k + 2) + k/(k + 1) = 1/(k + 1) [k(k + 2) +1]/(k + 2) = 1/(k + 1) [k² + 2k + 1]/(k + 2) =1/(k + 1) [(k + 1) (k + 1)]/(k + 2) = (k + 1)/(k + 2) P (n) is true for n = k + 1 Thus, P (n) is true for all n ∈ N.

Prove by the principle of mathematical induction:1/1.2 + 1/2.3 + 1/3.4 + … + 1/n(n + 1) = n/(n + 1)

Answer»

Suppose P (n) = 1/1.2 + 1/2.3 + 1/3.4 + … + 1/n(n+1) = n/(n+1)

For, n = 1

P (n) = 1/1.2 = 1/1+1

1/2 = 1/2

P (n) is true for n = 1

Now, let’s check for P (n) is true for n = k,

1/1.2 + 1/2.3 + 1/3.4 + … + 1/k(k+1) + k/(k+1) (k+2) = (k+1)/(k+2)

Now,

1/1.2 + 1/2.3 + 1/3.4 + … + 1/k(k + 1) + k/(k + 1) (k + 2)

= 1/(k + 1)/(k + 2) + k/(k + 1)

= 1/(k + 1) [k(k + 2) +1]/(k + 2)

= 1/(k + 1) [k² + 2k + 1]/(k + 2)

=1/(k + 1) [(k + 1) (k + 1)]/(k + 2)

= (k + 1)/(k + 2)

P (n) is true for n = k + 1

Thus, P (n) is true for all n ∈ N.