Binomial Theorem: Evaluating Complex Combinations

[ritwik06]

Homework Statement

Evaluate
$$\sum^{m}_{r=0} ^{ n + r }C_{n}$$

I can handle things when the lower thing in the combination part is changing, what shall I do with this one?

 

[rock.freak667]

Try writing out a few terms in the series and see if it helps.

 

[ritwik06]

I get this:
$$^{n}C_{n} + ^{n+1}C_{n} + ^{n+2}C_{n} + ... + ^{n+r}C_{n}$$
$$^{n}C_{0} + ^{n+1}C_{1} + ^{n+2}C_{2} + ... + ^{n+r}C_{r}$$

All I can do is this, now both the superscript and th subscript are increasing in A.P.

 

[ritwik06]

The thread is still unsolved...

 

[HallsofIvy]

the suggestion was that you actually look at a few specific examples.
If m= 1, you have
$$^nC_n+ ^{n+1}C_n= \frac{n!}{n!0!}+ \frac{(n+1)!}{n!1!}= 1+ n+ 1= n+ 2$$
If m= 2, you have
$$^nC_n+ ^{n+1}C_n+ ^{n+2}C_n= n+ 2+ \frac{(n+2)!}{n! 2!}= n+ 2+ (n+1)(n+2)/2= n+2+ \frac{1}{2}n^2+ \frac{3}{2}x+ 1= \frac{1}{2}n^2+ \frac{5}{2}n+ 3$$

Try a few more like that and see if anything comes to mind.