Nhị thức Newton.

Question

Tìm hệ số lớn nhất trong khai triển: $\left ( \frac{2010}{2012}+\frac{2x}{2013} \right )^{10}$

score 1 · Answer 1

Hệ số của

$x^k$ trong khai triển là:

$a_k=C_{10}^k\left(\dfrac{2}{2013}\right)^k\left(\dfrac{2010}{2012}\right)^{10-k}$

Ta sẽ chứng minh:

$a_k>a_{k+1},\forall k$

$\Leftrightarrow C_{10}^k\left(\dfrac{2}{2013}\right)^k\left(\dfrac{2010}{2012}\right)^{10-k}>C_{10}^{k+1}\left(\dfrac{2}{2013}\right)^{k+1}\left(\dfrac{2010}{2012}\right)^{9-k}$

$\Leftrightarrow \dfrac{10!}{k!(10-k)!}\left(\dfrac{2}{2013}\right)^k\left(\dfrac{2010}{2012}\right)^{10-k}>\dfrac{10!}{(k+1)!(9-k)!}\left(\dfrac{2}{2013}\right)^{k+1}\left(\dfrac{2010}{2012}\right)^{9-k}$