Final answer:
After analyzing the pattern in the sequence 1, -12, 14, -18, 116, it is determined that the absolute differences between numbers are doubling. However, the calculated next number based on this pattern, -152, does not match any of the given options. Therefore, either there is a mistake in the pattern detection, or the correct option is not provided.
Step-by-step explanation:
To predict the next number in the sequence (1, -12, 14, -18, 116), we must first determine the pattern. Analyzing the sequence, we notice that each number alternates between positive and negative, and the absolute value of each number appears to double the previous one. Starting from 1, the sequence of absolute values seems to be 1, 6, 7, 9, 58, which does not follow a simple doubling pattern. However, observing the pattern in the absolute values of differences, it reveals a pattern of doubling (5, 10, 15, 20, ...). Therefore, if we follow this pattern:
- The difference between the first two numbers is -12 minus 1 which is -13.
- The difference between the second and third numbers is 14 minus -12 which is 26.
- The difference between the third and fourth numbers is -18 minus 14 which is -32 (26 doubled would be 52).
- The difference between the fourth and fifth numbers is 116 minus -18 which is 134 (66 doubled, which is half of 132).
Following this pattern, the difference we expect next would be the previous difference 134 doubled, which is 268. Since the numbers alternate in sign, and the last number 116 is positive, the next number should be negative. Therefore, subtracting 268 from 116, we predict that the next number in the sequence is -152. The closest option given is -132, so we could have made an error in our pattern detection, or the provided options do not match our calculated result.
Since none of the given options exactly match our calculation, we have to revise our pattern recognition. Assuming that we incorrectly analyzed the pattern of differences and actually it's the sequence of the numbers doubled, the next difference after 134 would be 134 * 2 = 268, and the next term would be the last term minus this difference (because we alternate the sign), which is 116 - 268 = -152. Therefore, the correct answer using inductive reasoning that matches our calculation is not provided in the options A through D.