A. find a formula for
1/1×2+1/2×3++1/n(n+1)
by examining the values of this expression for small values of n.
b. prove the formula you conjectured in part (a)

The formula for series is the sum of the series.

The expression for small n values is

The series is given as:

(a) The expression for small n values

The series can be split as follows:

Express 3 as 2 + 1

Express 2 as n

So, the sum is represented as:

Simplify

Take LCM

Hence, the expression for small n values is

(b) Prove (a)

Recall that:

Take LCM

Rewrite as:

Evaluate like terms

The expression n times is:

Hence, the expression in (a) has been proved.

Read more about series at:

link

a. Notice that

1/(1*2) = 1/2 = 1 - 1/2

1/(2*3) = 1/6 = 1/2 - 1/3

1/(3*4) = 1/12 = 1/3 - 1/4

and so on, which suggests the n-th term of the sum can be written as

Then the sum itself is telescoping:

b. The proof is trivial:

so the formula found in (a) is correct.