Using the principle of mathematical induction, prove that `(1-1/2)(1-1/3)(1-1/4)...(1-1/(n+1))= 1/((n+1))" for all " n in N`.

Using the principle of mathematical induction, prove that `(1-1/2)(1-1

1.	Using the principle of mathematical induction, prove that `(1-1/2)(1-1/3)(1-1/4)...(1-1/(n+1))= 1/((n+1))" for all " n in N`.
Answer» Let the given statement be P(n). Then, `P(n) : (1-1/2)(1-1/3)(1-1/4)...(1-1/(n+1))= 1/((n+1))` When ` n = 1, LHS = (1-1/2) = 1/2 and RHS = 1/((1+1)) = 1/2`. ` :. ` LHS = RHS. Thus, P(1) is true. Let P(k) be true. Then, ` P(k): (1-1/2)(1-1/3)(1-1/4)...(1-1/(k+1)) = 1/((k+1))`. ...(i) Now, `{(1-1/2)(1-1/3)(1-1/4)...(1-1/(k+1))}(1-1/(k+2))` ` = 1/((k+1))[((k+2)-1)/((k+2))]=1/((k+1))*((k+1))/((k+2)) = 1/((k+2))` [using (i)]. ` :. P(k+1):(1-1/2)(1-1/3)(1-1/3)(1-1/4)...(1-1/(k+2))= 1/((k+2))`. This shows that P(k+1) is true, whenever P(k) is true. Thus, P(1) is true and P(k+1) is true, whenever P(k) is true. Hence , by the principle of mathematical induction, we have `(1-1/2)(1-1/3)(1-1/4)...(1-1/(n+1)) = 1/((n+1)) " for all " n in N`.