2).Find the sum to n terms of 1/(1.2.3) + 3/(2.3.4) + 5/(3.4.5) + 7/(4.5.6)

To find the sum of the given series, we can first observe that the terms in the series have a pattern. Let's look at each term separately:

1st term: 1/(1.2.3)
2nd term: 3/(2.3.4) = 1/(2.3.4) * 3
3rd term: 5/(3.4.5) = 1/(2.3.4) * 5
4th term: 7/(4.5.6) = 1/(2.3.4) * 7

We can see that the denominator of each term has the same pattern: 2.3.4, which we can rewrite as (2 + 1).(3 + 1).(4 + 1).

So, each term can be written as 1/(2.3.4) multiplied by the numerator of the term.

Now, let's write the general term, Tn, in the series:

Tn = (2n - 1)/[(2 + 1).(3 + 1).(4 + 1)] = (2n - 1)/(n.(n + 1).(n + 2))

To find the sum of the series up to n terms, Sn, we need to add all the terms from the first term (n = 1) to the nth term.

Sn = T1 + T2 + T3 + ... + Tn

Substituting the value of Tn into the sum expression:

Sn = [(2*1 - 1)/(1*(1 + 1)*(1 + 2))] + [(2*2 - 1)/(2*(2 + 1)*(2 + 2))] + [(2*3 - 1)/(3*(3 + 1)*(3 + 2))] + ... + [(2n - 1)/(n*(n + 1)*(n + 2))]

Simplifying this expression, we get the sum of the series to n terms.