Find the first term and the common ratio for the geometric sequence. Round approximations to the nearest hundredth.
-$$a _ { 2 } = - 78 , a _ { 5 } = 624$$
A) $\mathrm { a } 1 = 39 , \mathrm { r } = - 2$
B) $\mathrm { a } _ { 1 } = 39 , \mathrm { r } = - 0.5$
C) $\mathrm { a } _ { 1 } = 624 , \mathrm { r } = - 0.5$
D) $a _ { 1 } = - 78 , r = - 2$

Question

Quizplus · Accepted Answer

Given $a_2 = -78$ and $a_5 = 624$, we can use the formula for the nth term of a geometric sequence, $a_n = a_1 \cdot r^{(n-1)}$, to find the first term ($a_1$) and the common ratio ($r$). For $a_2$, we have $-78 = a_1 \cdot r^{(2-1)} = a_1 \cdot r$, and for $a_5$, we have $624 = a_1 \cdot r^{(5-1)} = a_1 \cdot r^4$. Dividing the second equation by the first gives us $r^3 = -8$, so $r = -2$. Substituting $r = -2$ into the equation for $a_2$, we find $a_1 = 39$.

Find the First Term and the Common Ratio for the Geometric