Question

prove by mathematical induction.........,................................1/1.2 + 1/2.3 +-----+1/n(n+1) (n+2) = n(n+3)/4(n+1) (n+2)

Answer 1

Explanation:

Induction method is used to prove a statement. Most commonly, it is used to prove a statement, involving, say n where n represents the set of all natural numbers.

Induction method involves two steps, One, that the statement is true for n=1 and say n=2. Two, we assume that it is true for n=k and prove that if it is true for n=k, then it is also true for n=k+1.

First Step − Now for 1/1⋅2+1/2⋅3+...+1/n(n+1)=n/n+1, we know for n=1, we have 1/1⋅2=1/2 and for n=2, we have 1/1⋅2+1/2⋅3=1/2+1/6=2/3=2/2+1.

Hence, given statement is true for n=1 and n=2.

Second Step − Assume it is true for n=k, hence

1/1⋅2+1/2⋅3+...+1/k(k+1)=k/k+1

Now let us test it for n=k+1 i.e.

1/1⋅2+1/2⋅3+...+1/k(k+1)+1/(k+1)(k+2)

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

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

= $k^{2}$+2k+1/(k+1)(k+2)

= $(k+1)^{2}$/(k+1)(k+2)

= k+1/k+2

Hence we see that the statement is true for n=k+1 if it is true for n=k.

Hence 1/1⋅2+1/2⋅3+...+1/n(n+1)=n/n+1