find the common ratio of the geometric progression ¼ ½ 1—2
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The solution
To find the common ratio of a geometric progression, we need to observe the pattern in the sequence of terms and determine the ratio between consecutive terms.
Let's look at the given sequence: ¼, ½, 1, -2
To find the common ratio, we need to divide each term by its preceding term.
The common ratio (r) can be found as follows:
r = ½ ÷ ¼ = 2/4 = 1/2 (ratio between the 2nd and 1st terms)
r = 1 ÷ ½ = 2/1 = 2 (ratio between the 3rd and 2nd terms)
r = -2 ÷ 1 = -2 (ratio between the 4th and 3rd terms)
Therefore, we have multiple ratios in this sequence: 1/2, 2, and -2. This means the given sequence is not a geometric progression because it does not have a constant common ratio.
Please note that for a sequence to be a geometric progression, all the terms must have the same common ratio. In this case, ¼, ½, 1, -2 do not have a consistent ratio, hence it cannot be considered a geometric progression.