To find the missing terms in the geometric sequence that includes the terms 3072 and 12, we first need to identify the common ratio of the sequence.
In a geometric sequence, each term can be found by multiplying the previous term by a constant ratio, denoted as \( r \). So, if we denote the first term before 3072 as \( a \) and the second term after 3072 as \( b \), the terms can be related as follows:
- The term before 3072 will be \( a \).
- The term after 3072 will be \( 3072 \cdot r \).
- The term after that will be \( 3072 \cdot r^2 \).
- The term after that will be \( 3072 \cdot r^3 \).
- Finally, we know the last term in the subset is \( 12 \).
Since we know that the sixth term (counting from 3072) is 12, we can set up the equation:
\[ 12 = 3072 \cdot r^n \]
where \( n \) is the number of steps from 3072 to 12. To find \( n \):
- From 3072 to the 12, we have 4 steps, so \( n = 4 \).
Then we can express \( r^4 \):
\[ r^4 = \frac{12}{3072} \]
Calculating this gives:
\[ r^4 = \frac{1}{256} \]
Taking the fourth root of both sides:
\[ r = \frac{1}{4} \]
Now we can backtrack to find the terms preceding 3072:
- First term before 3072 (by dividing by \( r \)): \[ a = 3072 \cdot \frac{1}{4} = 768 \]
- The term before 768: \[ = 768 \cdot \frac{1}{4} = 192 \]
- The term before 192: \[ = 192 \cdot \frac{1}{4} = 48 \]
Now we have established the missing terms preceding 3072 as 768, 192, and 48.
Thus, the complete sequence looks like:
... , 768, 192, 48, 3072, 12, ...
The correct response is:
768, 192, 48
1 answer