Bacteria quadruples on a daily basis. After the first day, there were 50 bacteria. Using the general exponential equation an=a1⋅rn−1

Question

Bacteria quadruples on a daily basis. After the first day, there were 50 bacteria. Using the general exponential equation  an=a1⋅rn−1
 , what value would represent  a1
 ?
The value of a1
 is

GPT-4o mini · Answer

In the given exponential equation $ a_n = a_1 \cdot r^{n-1} $, the term $ a_1 $ represents the initial quantity (the starting amount) of the bacteria at $ n = 1 $. Given that the bacteria quadruples (increases by a factor of 4) daily, we can denote the growth factor $ r $ as 4. From the problem, we know that after the first day ($ n = 2 $), the quantity of bacteria is 50. Thus: $$ a_2 = a_1 \cdot r^{2-1} = a_1 \cdot r $$ Substituting the values we know: $$ 50 = a_1 \cdot 4 $$ Now we can solve for $ a_1 $: $$ a_1 = \frac{50}{4} = 12.5 $$ So, the value of $ a_1 $ is $ 12.5 $.