a scientist has discovered an organism that produces five offspring exactly one hour after its own birth…

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?\n2,801\n19,531\n19,607\n97,655

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?\n2,801\n19,531\n19,607\n97,655

Answer

Explanation:

Step1: Identify the pattern

This is a geometric - sequence problem. The initial number of organisms (a_1 = 1). After 1 hour, the number of new - born organisms is 5, so the total number of organisms (a_2=1 + 5=6). The common ratio (r = 6) (because each existing organism produces 5 offspring, so the total number of organisms at each step is 6 times the number of organisms at the previous step).

Step2: Determine the number of time - steps

We want to find the number of organisms at 7 hours. The number of time - steps (n=7) (starting from (n = 1) at the first hour).

Step3: Use the geometric - sequence formula

The formula for the (n)th term of a geometric sequence is (a_n=a_1r^{n - 1}). Here, (a_1 = 1), (r = 6), and (n = 7). Substitute these values into the formula: (a_7=1\times6^{7 - 1}=6^6).

Step4: Calculate (6^6)

(6^6=6\times6\times6\times6\times6\times6 = 46656). But this is wrong. Let's consider the correct way. We can build the sequence step - by - step. At (t = 0), (N_0=1). At (t = 1), (N_1=1\times5 + 1=6). At (t = 2), (N_2=6\times5+6 = 6\times(5 + 1)=6^2). In general, at (t = n), (N_n=6^n). For (n = 7), (N_7=6^7=279936). There is a mistake above. Let's start over. We know that: At (t = 0), there is 1 organism. At (t = 1), the original organism produces 5 offspring, so there are (1+5=6) organisms. At (t = 2), each of the 6 organisms from (t = 1) produces 5 offspring, so the number of new organisms is (6\times5), and the total number of organisms is (6+6\times5=6\times(1 + 5)=6^2). By induction, at (t = n) hours, the total number of organisms (T(n)=6^n). When (n = 7), (T(7)=6^7=279936). There is still a problem. Let's consider the following: At (t = 0), (O_0 = 1). At (t=1), (O_1=5) (offspring of the initial organism). Total (=1 + 5=6). At (t = 2), the 5 organisms from (t = 1) produce (5\times5) offspring, and the initial organism and its first - generation offspring still exist. So the total number of organisms (O_2=1+5 + 5^2). At (t = 3), the organisms from (t = 2) produce offspring. The total number of organisms (O_3=1+5 + 5^2+5^3). In general, the sum of a geometric series (S_n=\sum_{k = 0}^{n}a\times r^k=\frac{a(r^{n + 1}-1)}{r - 1}) (here (a = 1), (r = 5), and we want to find the number of organisms at (t = 7), so (n=7)). (S_7=\frac{5^{8}-1}{5 - 1}=\frac{390625-1}{4}=\frac{390624}{4}=97656).

Answer:

97656