Question

How do you find the 7th term in the binomial expansion for `(x - y)^6`?

Answer 1

`T_7=y^6`

Explanation:

From the Binomial Theorem, we obtain that for the expression
`(x+y)^n=sum_(r=0)^n""^nC_rx^(n-r)y^r`,

The term in position `(r+1)` is given by

`T_(r+1)=""^nC_rx^(n-r)y^r`

Hence the 7th term is the term in position `T_(6+1)` and may be given by

`T_7=T_(6+1)=""^6C_6x^(6-6)(-y)^6`

`=y^6`

Alternatively, we also note that any binomial of form `(x+y)^n` always has `(n+1)` terms.
Hence the 7th term is the last term of the series `sum_(r=0)^n""^nC_rx^(n-r)y^r`, and is hence `""^6C_6x^(6-6)(-y)^6=y^6`