Find the first four terms of the binomial series for the given function.
-$$\left( 1 + 9 x ^ { 2 } \right) ^ { 3 }$$
A) $1 + 27 x ^ { 2 } + 27 x ^ { 4 } + x ^ { 6 }$
B) $1 + 27 x ^ { 2 } + 405 x ^ { 4 } + 5103 x ^ { 6 }$
C) $1 + 27 x ^ { 2 } + 243 x ^ { 4 } + 729 x ^ { 6 }$
D) $1 + 27 x ^ { 2 } + 81 x ^ { 4 } + 243 x ^ { 6 }$

Question

Quizplus · Accepted Answer

The binomial series for $\left(1+aight)^n$ is $$1+n a+\frac{n\left(n-1ight)}{2!}a^2+\frac{n\left(n-1ight)\left(n-2ight)}{3!}a^3+\cdots$$
In this case, we have $a=9x^2$ and $n=3.$ Applying the formula, we get the first four terms as follows: $$1+3\cdot 9x^2+\frac{3\cdot 2}{2!}\cdot \left(9x^2ight)^2+\frac{3\cdot 2\cdot 1}{3!}\cdot\left(9x^2ight)^3=1+27x^2+243x^4+729x^6,$$ which corresponds to $\boxed{	extbf{(C)}}.$

Find the First Four Terms of the Binomial Series for the Given