Binomial coefficients

[Albi]

Guys, can someone explain to me why the expression below holds, please try to explain it as simple as possible, thanks in advance.[math]\sum^ {k+1} _{i=0}\binom{n+k+1}{k+1}= \left ( \sum ^{k}_{i=0}\binom{n+k}{k} \right )+ \binom{n+k+1}{k+1}[/math]

 

[lex]

It only holds when n=0.
I suspect it is not written down correctly.

 

[Dr.Peterson]

Guys, can someone explain to me why the expression below holds, please try to explain it as simple as possible, thanks in advance.[math]\sum^ {k+1} _{i=0}\binom{n+k+1}{k+1}= \left ( \sum ^{k}_{i=0}\binom{n+k}{k} \right )+ \binom{n+k+1}{k+1}[/math]

There ought at least to be an i in the addends!

Can you show the context in the original source?

 

[Albi]

Guys, can someone explain to me why the expression below holds, please try to explain it as simple as possible, thanks in advance.[math]\sum^ {k+1} _{i=0}\binom{n+k+1}{k+1}= \left ( \sum ^{k}_{i=0}\binom{n+k}{k} \right )+ \binom{n+k+1}{k+1}[/math]

I think this is the correct one [math]\sum^ {r+1} _{k=0}\binom{n+k}{k}= \left ( \sum ^{r}_{k=0}\binom{n+k}{k} \right )+ \binom{n+r+1}{r+1}[/math]

 

[Steven G]

What is the difference between the left hand side and the 1st term on the right hand side. What must you add to the the 1st term on the right hand side to get the left hand side?

 

[Albi]

What is the difference between the left hand side and the 1st term on the right hand side. What must you add to the the 1st term on the right hand side to get the left hand side?

There is one more term on the left side, I think I must add add [imath]\binom{n+k+1}{k}[/imath], but I'm not sure tho.

 

[Dr.Peterson]

I think this is the correct one [math]\sum^ {r+1} _{k=0}\binom{n+k}{k}= \left ( \sum ^{r}_{k=0}\binom{n+k}{k} \right )+ \binom{n+r+1}{r+1}[/math]

There is one more term on the left side, I think I must add add [imath]\binom{n+k+1}{k}[/imath], but I'm not sure tho.

The sum on the right is all but the last term of the sum on the left. So you have to add to it the last term of the latter, which is what you get when you replace k in [imath]\binom{n+k}{k}[/imath] with r+1. What do you get when you do that? [imath]\binom{n+r+1}{r+1}[/imath], which is the second term on the right.

 

[Albi]

The sum on the right is all but the last term of the sum on the left. So you have to add to it the last term of the latter, which is what you get when you replace k in [imath]\binom{n+k}{k}[/imath] with r+1. What do you get when you do that? [imath]\binom{n+r+1}{r+1}[/imath], which is the second term on the right.

Thanks!