# Sequences and series problem

1. Dec 17, 2012

### Vineeth T

1. The problem statement, all variables and given/known data

If P r=(n-r)(n-r+1)(n-r+2).....(n-r+p-1)
Qr= r(r+1)(r+2)......(r+q-1)
Find P1Q1+P2Q2+.......
+Pn-1Qn-1

2. Relevant equations

3. The attempt at a solution
I tried to bring the general term in the form of a coefficient of x in the binomial expansion.
But it does not simplify to that form.
can anyone give me a better way to approach the problem?

2. Dec 17, 2012

### Bonaparte

Its a lot of messing around with, but I'll give you the basic idea, note that when you expand, the integers turn out to be sum of squares, that is (n-1)(n)(2n-1)/6. The rn turn out to be (n-1)rn. You just do this to the different terms to get the final thing, which then you might be able to factor out. Try doing this and post your result, shouldn't be too hard.

Thanks, Bonaparte

3. Dec 17, 2012

### Vineeth T

Can you explain it more clearly?
Also the source of this problem is from a book called "HIGHER ALGEBRA" by Hall&Knight.
If you have this book see the answer (only the final result is given) in pg:328.Q no:27
The answer even has factorial notations in it.

Last edited: Dec 17, 2012
4. Dec 23, 2012

### sankalpmittal

Try evaluating P1 , P2 , .... , Pn-1 .....

Then Q1 , Q2 , Q3 , ..... , Qn-1...

Don't over simplify....

Then You evaluate P1Q1+P2Q2+.......
+Pn-1Qn-1...

First do this much. What do you get ?