How Can the Method of Differences Be Applied to Solve This Series Problem?

[crays]

Hi, I have this question that suppose to use Method of Differences to solve it.

By using the method of differences, find the sum of the first n terms of the series whose r^th term, u_r, are

U_r = (2r - 1)/r(r+1)(r+2)

I used partial fraction and found.

U_r = -1/(2r) + 3/(r+1) - 5/[2(r+2)]

then i did

-(1/2)[5/(r+2) - 6/(r+1)] - 1/(2r)
-(1/2)[5/(n+2) - 5/2] - (1/2)(1/r)
5/4 - 5/(2n+4) - (1/2)[1/(1/2)(n)(n+1)]
(10n + 20 - 20)/4(2n+4) - 1/(n)(n+1)

But the answer is wrong. Any help?
Thanks.

 

[fundoo]

You have made wrong partial fractions. that's not the way.

first separate the terms in numerator so U_r = 2/(r+1)(r+2) - 1/r(r+1)(r+2)

and now splitting them gives

U_r = 2(1/(r+1) - 1/(R+2)) - 1/2 (1/r(r+1) - 1/(r+1)(r+2)

so the final Answer will be = 2(1/2 - 1/(r+2)) - 1/2 (1/2 - 1/(r+1)(r+2))