In a certain list of numbers, the first number is 2, the second number is 3, and each succeeding number is the sum of all the numbers that precede it in the list. If hand k denote the 20th and 24th numbers in the list, respectively, what is the value of k/h?
k/h equals 16.
In the given sequence, each number is derived from the sum of all preceding numbers, leading to exponential growth. By calculating the 20th and 24th terms, we find their ratio to equal 16.
The ratio k/h equating to 5 would imply that the 20th number is significantly smaller than the 24th number; however, the growth of the sequence results in much larger values, making this choice inconsistent with the calculated values.
Selecting 8 as the ratio would mean that the 24th number is eight times the 20th number. Given the exponential growth of the sequence, this ratio does not hold true, as the values escalate much more rapidly.
A ratio of 10 suggests that the 24th number is ten times the 20th. However, the actual numbers in the sequence grow at a rate that does not support such a ratio, as the terms increase dramatically due to their recursive nature.
This choice indicates that the 24th number is 16 times larger than the 20th number. Upon calculating the specific terms in the sequence, this ratio is confirmed, making it the correct solution.
If k/h were equal to 20, it would imply an even larger discrepancy between the 20th and 24th numbers than what is present. The rapid growth of the sequence rules out this option as well.
The sequence grows exponentially due to its recursive definition, leading to the conclusion that the ratio of the 20th and 24th numbers is 16. This ratio reflects the significant increase in value as more terms are added, ultimately demonstrating the unique nature of this sequence.
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