Question

What is the seventh term of (x + 4)8?

A. 114,688x2
B. 114,688x3
C. 114,688x4
D. 114,688x5

Answer 1

The (p+1)-th term of the Newton binomial expansion


`(a+b)^(n)`

is given by


`t_(p+1)=\dbinom{n}{p}\,a^(n-p)\,b^(p)`
__________________________

We want the 7th term. Hence, we set
`p+1` to be
`7:`


`p+1=7~~\Rightarrow~~p=6`


Then, the 7th term is


`t_(7)=\dbinom{8}{6}\,x^(8-6)\,4^(6)\\\\\\ t_(7)=(8!)/(6!\cdot (8-6)!)\cdot x^(8-6)\cdot 4^(6)\\\\\\ t_(7)=(8\cdot 7\cdot \diagup\!\!\!\! 6!)/(\diagup\!\!\!\! 6!\cdot 2!)\cdot x^(2)\cdot 4^(6)\\\\\\ t_(7)=(8\cdot 7)/(2\cdot 1)\cdot x^(2)\cdot 4^(6)\\\\\\ t_(7)=28\cdot x^(2)\cdot 4,096\\\\ \boxed{\begin{array}{c} t_(7)=114,688\,x^(2) \end{array}}`


Correct answer:
`\text{A. }114,688\,x^(2).`