Question

name a10 hzo jones date 01/23/ lesson 9: exit ticket geometric meanies find the missing terms in the following geometric sequence: 12, _, _, ___, 3,072

Explanation:

Step1: Find the common ratio

The formula for the $n$th term of a geometric sequence is $a_n=a_1r^{n - 1}$, where $a_n$ is the $n$th term, $a_1$ is the first - term, $r$ is the common ratio, and $n$ is the term number. Here, $a_1 = 12$, $n=5$, and $a_5 = 3072$. Substitute into the formula: $a_5=a_1r^{5 - 1}$, so $3072 = 12r^{4}$. Then $r^{4}=\frac{3072}{12}=256$. Taking the fourth - root of both sides, $r=\pm4$.

Step2: Find the missing terms when $r = 4$

For the second term ($n = 2$), $a_2=a_1r=12\times4 = 48$.
For the third term ($n = 3$), $a_3=a_1r^{2}=12\times4^{2}=12\times16 = 192$.
For the fourth term ($n = 4$), $a_4=a_1r^{3}=12\times4^{3}=12\times64 = 768$.

Step3: Find the missing terms when $r=-4$

For the second term ($n = 2$), $a_2=a_1r=12\times(-4)=-48$.
For the third term ($n = 3$), $a_3=a_1r^{2}=12\times(-4)^{2}=12\times16 = 192$.
For the fourth term ($n = 4$), $a_4=a_1r^{3}=12\times(-4)^{3}=12\times(-64)=-768$.

Answer:

If $r = 4$: 48, 192, 768; If $r=-4$: - 48, 192, - 768