Question

a scientist has discovered an organism that produces five offspring exactly one hour after its own birth, and then goes on to live for one week without producing any additional offspring. each replicated organism also replicates at the same rate. at hour one, there is one organism. at hour two, there are five more organisms. how many total organisms are there at hour seven? 1082 19,531 97,655

Explanation:

Step1: Analyze initial and growth pattern

At hour 1, there is 1 organism. At hour 2, there are 1 + 5=6 organisms. The number of organisms at each hour forms a sequence where the number of new - born organisms is 5 times the number of organisms in the previous hour. This is a geometric - sequence problem with the general formula \(a_n=a_1r^{n - 1}\), where \(a_1\) is the first - term, \(r\) is the common ratio, and \(n\) is the number of terms. Here, \(a_1 = 1\) (the number of organisms at hour 1), \(r = 6\) (because the total number of organisms at each step is 6 times the number of organisms at the previous step: 1 original + 5 new = 6 times the previous amount), and \(n=7\) (since we want to find the number of organisms at hour 7).

Step2: Apply the geometric - sequence formula

The formula for the \(n\)th term of a geometric sequence is \(a_n=a_1r^{n - 1}\). Substituting \(a_1 = 1\), \(r = 6\), and \(n = 7\) into the formula, we get \(a_7=1\times6^{7 - 1}\).
\[a_7=6^6\]
\[a_7 = 46656\]

Answer:

46656