High School Mathematics - 2 7.4 General Term (T_{r+1})
Suppose we have to find the (r+1)^{th} term in the expansion of (x+y)^{n}
In its expansion we find that
T_{3} = ^{n}C_{2} x^{n-2}y^{2}
T_{4} = ^{n}C_{3} x^{n-3}y^{3}
We can see that all these terms have the following properties.
The suffix of C is one less than the number of terms.
The power of the first factor x is equal to the difference of the upper and lower suffixes of C.
The power of the second factor y and the suffix of C are same.
The sum of the power of x and y is equal to the index of the given binomial.
Hence (r+1)^{th} term is given by T_{r+1} = ^{n}C_{r}x^{n-r}y^{r}
Example: Find the tenth term in the expansion of (2x-y)^{11}. Solution: T_{10} = T_{9+1} = ^{11}C_{9}(2x)^{11-9}(-y)^{9} = ^{11}C_{2}(2x)^{2}(-y)^{9} Answer: -220x^{2}y^{9}