The ratio of the coefficient of x^15 to the term independent of x in the expansion of (x^2…

Question

The ratio of the coefficient of $x^{15}$ to the term independent of $x$ in the expansion of $\left(x^2+\dfrac2x\right)^{15}$ is:

Accepted Answer

$$T_{r+1}=\binom{15}{r}(x^2)^{15-r}(2x^{-1})^r=\binom{15}{r}2^r\,x^{30-3r}$$

Independent term: $30-3r=0\Rightarrow r=10$: $T_{11}=\binom{15}{10}2^{10}$

Term with $x^{15}$: $30-3r=15\Rightarrow r=5$: coefficient $=\binom{15}{5}2^5$

$$\text{Ratio}=\frac{\binom{15}{5}2^5}{\binom{15}{10}2^{10}}=\frac{1}{2^5}=\frac{1}{32}$$

Solutions · 1