Binomial expansion

• Aug 8th 2009, 07:40 PM
gtaplayr
Binomial expansion
From the binomial expansion theorem there are terms in
$\displaystyle x^{k+1}$,$\displaystyle x^ky$, ...... , $\displaystyle xy^k$,$\displaystyle y^{k+1}$.
What are the coefficients of the general term $\displaystyle x^ry^{k+1-r}$ for:
i) r = 0
ii) r = 1,.....,k
iii) r = k + 1
• Aug 8th 2009, 11:33 PM
CaptainBlack
Quote:

Originally Posted by gtaplayr
From the binomial expansion theorem there are terms in
$\displaystyle x^{k+1}$,$\displaystyle x^ky$, ...... , $\displaystyle xy^k$,$\displaystyle y^{k+1}$.
What are the coefficients of the general term $\displaystyle x^ry^{k+1-r}$ for:
i) r = 0
ii) r = 1,.....,k
iii) r = k + 1

See the Mathworld or the Wikipedia Article

CB
• Aug 8th 2009, 11:42 PM
Jhevon
Quote:

Originally Posted by gtaplayr
From the binomial expansion theorem there are terms in
$\displaystyle x^{k+1}$,$\displaystyle x^ky$, ...... , $\displaystyle xy^k$,$\displaystyle y^{k+1}$.
What are the coefficients of the general term $\displaystyle x^ry^{k+1-r}$ for:
i) r = 0
ii) r = 1,.....,k
iii) r = k + 1

It seems you are using the definition: $\displaystyle (x + y)^{k + 1} = \sum_{r = 0}^{k + 1} {k + 1 \choose r}x^ry^{k + 1 - r}$.

now just plug in the respective $\displaystyle r$'s and you will be able to find the coefficients.