Question

Write the fifteenth term of the binomial expansion of (a^2+b)^20

Answer 1

The fifteenth term of the binomial expansion of
`(a+b)^(20)` is
`38760\cdot a^(6)\cdot b^(14)`.

Explanation:

Let be a binomial of the form
`(a+b)^(n)`, where
`a, b\in \mathbb{R}` and
`n\,\in\mathbb{N}^(+)`. The expansion of this polynomial is defined below:

`(a+b)^(n) = \Sigma\limits_(k=0)^(n)\,(n!)/(k!\cdot (n-k)!)\cdot (a^(n-k)\cdot b^(k))` (1)

Where:

`n` - Number of terms of the expanded polynomial.

`k` - Index associated to
`k`-th term of the expanded polynomial.

For all
`n`-th binomial, we a sum of
`n+1` terms. If the given binomial has a term of
`20`, then we have 21 terms and the fifteenth term of the polynomial corresponds to the
`14`-th term. Then, the fifteenth term of the binomial is:

`c_(14) = (20!)/(14!\cdot 6!)\cdot (a^(6)\cdot b^(14))`

`c_(14) = 38760\cdot a^(6)\cdot b^(14)`

The fifteenth term of the binomial expansion of
`(a+b)^(20)` is
`38760\cdot a^(6)\cdot b^(14)`.