To find the common ratio of a geometric progression, we use the property that each term is obtained by multiplying the previous term by a constant factor known as the common ratio.
Given the first four terms of the progression: [tex]\(84, 42, 21, 10.5\)[/tex],
Let's denote the common ratio by [tex]\(r\)[/tex].
We can find [tex]\(r\)[/tex] by dividing the second term by the first term:
[tex]\[ r = \frac{42}{84} = 0.5 \][/tex]
To ensure that this is indeed the common ratio, we can also check the subsequent terms:
1. Divide the third term by the second term:
[tex]\[ r = \frac{21}{42} = 0.5 \][/tex]
2. Divide the fourth term by the third term:
[tex]\[ r = \frac{10.5}{21} = 0.5 \][/tex]
Since the ratio remains consistent across all terms, we can confirm that the common ratio of the geometric progression is:
[tex]\[ r = 0.5 \][/tex]
So, the common ratio of the progression is [tex]\(0.5\)[/tex].
