Question

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

Answer 1

Answer:

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

Step-by-step 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:

$(a+b)^{n} = \Sigma\limits_{k=0}^{n}\,\frac{n!}{k!\cdot (n-k)!}\cdot (a^{n-k}\cdot b^{k})$ (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:

$c_{14} = \frac{20!}{14!\cdot 6!}\cdot (a^{6}\cdot b^{14})$

$c_{14} = 38760\cdot a^{6}\cdot b^{14}$

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