Use Factorial Notation
Write the first four terms of the sequence whose general term is given.
-$$a _ { n } = \frac { 3 ^ { n } } { ( n + 3 ) ! }$$
A) $\frac { 1 } { 8 } , \frac { 3 } { 40 } , \frac { 3 } { 80 } , \frac { 9 } { 560 }$
B) $\frac { 3 } { 7 } , \frac { 9 } { 8 } , 3 , \frac { 81 } { 10 }$
C) $\frac { 4 } { 3 } , \frac { 9 } { 5 } , \frac { 9 } { 2 } , \frac { 7 } { 81 }$
D) $\frac { 1 } { 8 } , \frac { 3 } { 40 } , \frac { 3 } { 40 } , \frac { 9 } { 280 }$

Question

Quizplus · Accepted Answer

We have $a_n = \frac{3^n}{(n+3)!}$. To write this in factorial notation, we can write it as $a_n = \frac{3^n}{(n+3)(n+2)(n+1)n!} = \frac{3}{(n+3)(n+2)(n+1)}\cdot\frac{3^{n-3}}{(n-3)!}$. The first four terms of this sequence are then $\frac{3}{6\cdot5\cdot4}\cdot\frac{3^0}{0!} = \frac{1}{40}$, $\frac{3}{7\cdot6\cdot5}\cdot\frac{3^1}{1!} = \frac{3}{140}$, $\frac{3}{8\cdot7\cdot6}\cdot\frac{3^2}{2!} = \frac{3}{560}$, and $\frac{3}{9\cdot8\cdot7}\cdot\frac{3^3}{3!} = \frac{9}{560}$. This matches the first four terms in choice $\boxed{	extbf{(A)}}$.

Use Factorial Notation Write the First Four Terms of the Sequence

Use Factorial Notation
Write the First Four Terms of the Sequence