Question

What is the fifth term of `(x + y)^8`?

Answer 1

Use the formula `t_(r + 1) = _nC_r(a)^(n - r)(b)^r` to determine the `rth` term of the expansion.

Explanation:

Since we're looking for the 5th term, r = 4. The value of n is the exponent. "a" is the base to the left and "b" the one to the right.

`t_5 =_8C_4(x)^(8 -4)(y) ^4`

Calculating `""_8C_4` using the combination formula:

`""_nC_r` = `(n!) / ((n - r)!r!)`

`""_8C_4` = `(8!)/((8 - 4)!4!)`

`""_8C_4` = `(8(7)(6)(5)(4)(3)(2)(1))/(((4)(3)(2)(1))(4)(3)(2)(1))`

After simplifying:

`""_8C_4` = `1680/24`

`""_8C_4` = 70

`t_5 = 70(x^4)(y^4)`

`t_5 = 70x^4y^4`

So, term 5 has a value of `70x^4y^4`

Practice exercises:

1. Find the middle term of `(2x - 3y)^12`

2. Which term in `(4x + y)^11` would be constant?