Mathematics - Binomial Theorem Question with Solution | TestHub

Since $(x^2+2x+4{)}^n={\sum }_{r=0}^{2n}{a}_r{x}^r$, coefficients are symmetric, $a_r=a_{2n-r}$, so $\frac{a_{2n-r}}{a_r}=1$. The sum becomes ${\sum }_{r=n}^{2n}1=2n-n+1=n+1$. Using the concept of geometric progression, the final answer is $\frac{4^{n+1}-1}{3}$.

Final Answer: The final answer is $\boxed{\frac{4^{\mathrm{n}+1}-1}{3}}$