Find the first term and the common ratio for the geometric sequence. Round approximations to the nearest hundredth.
-$$a _ { 2 } = 9 , a _ { 6 } = 729$$
A) $\mathrm { a } _ { 1 } = 3 , \mathrm { r } = 0.33$
B) $\mathrm { a } _ { 1 } = 3 , \mathrm { r } = 3$
C) $\mathrm { a } _ { 1 } = 729 , \mathrm { r } = 0.33$
D) $a _ { 1 } = 9 , r = 3$

Question

Quizplus · Accepted Answer

Given $a_2 = 9$ and $a_6 = 729$, we can use the formula for the $n$th term of a geometric sequence, $a_n = a_1 \cdot r^{(n-1)}$, to find the first term $a_1$ and the common ratio $r$. From $a_2 = 9$, we have $a_1 \cdot r^{(2-1)} = 9$, which implies $a_1 \cdot r = 9$. From $a_6 = 729$, we have $a_1 \cdot r^{(6-1)} = 729$, which implies $a_1 \cdot r^5 = 729$. Dividing the second equation by the first gives $r^4 = 81$, so $r = 3$. Substituting $r = 3$ into $a_1 \cdot r = 9$ gives $a_1 = 3$.

Find the First Term and the Common Ratio for the Geometric