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? 2,801 19,531 19,607 97,655

Explanation:

Step1: Identify the growth pattern

This is a geometric - sequence problem. The first - term \(a_1 = 1\), and the common ratio \(r = 5\). The formula for the sum of a geometric sequence is \(S_n=\frac{a_1(r^n - 1)}{r - 1}\) when \(r
eq1\).

Step2: Determine the values of \(n\), \(a_1\), and \(r\)

We want to find the total number of organisms at hour seven. The number of terms \(n = 7\), \(a_1=1\), and \(r = 5\).

Step3: Substitute the values into the formula

\[S_7=\frac{1\times(5^7 - 1)}{5 - 1}=\frac{5^7-1}{4}\]
\[5^7=5\times5\times5\times5\times5\times5\times5 = 78125\]
\[S_7=\frac{78125 - 1}{4}=\frac{78124}{4}=19531\]

Answer:

19,531