Final answer:
The common ratio for the geometric progression 1/4, -1/2, 1, -2 is -2, which is determined by dividing any term in the sequence by the previous term while considering the rules of multiplication for positive and negative numbers.
Step-by-step explanation:
The common ratios of the given geometric sequence 1/4, -1/2, 1, -2 can be found by dividing any term by the previous term. Firstly, to find the ratio between -1/2 and 1/4, you divide -1/2 by 1/4, which is the same as multiplying -1/2 by the reciprocal of 1/4. This gives us a common ratio of -2. Then, we find the ratio between 1 and -1/2 by dividing 1 by -1/2, which again, is the same as multiplying 1 by the reciprocal of -1/2, resulting in a common ratio of -2. Lastly, we divide -2 by 1 to find the common ratio between -2 and 1, which comes out to be -2. Therefore, the common ratio for this sequence is -2.
The properties of multiplication involving positive and negative numbers are very important when determining these ratios. Multiplication rules state that the product of two positive numbers is positive, just as the product of two negative numbers is also positive. Conversely, when the numbers multiplied have opposite signs, the result is negative [math terms].