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

1 Answer
Oct 24, 2015

#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#