Let [t] denote the greatest integer t.

Question

Let $[t]$ denote the greatest integer $\le t$. If the constant term in the expansion of $\left(3x^2-\dfrac{1}{2x^5}\right)^7$ is $\alpha$, then $[\alpha]$ is equal to \_\_\_\_\_\_\_.

Answer: 1275

Accepted Answer

$$T_{r+1}=\binom7r(3x^2)^{7-r}\left(-\frac{1}{2x^5}\right)^r$$

Power: $14-2r-5r=14-7r=0\Rightarrow r=2$.

$$T_3=\binom72\,3^5\left(\frac12\right)^2=21\times\frac{243}{4}=1275.75$$

$$[\alpha]=1275$$

Solutions · 1