Solve each problem . (Modeling) Bacterial Growth The strain...

Solve each problem . (Modeling) Bacterial Growth The strain of bacteria described in Exercise 85 in the first section of this chapter will double in size and then divide every 40 minutes. Let $a_{1}$ be the initial number of bacteria cells, $a_{2}$ the number after 40 minutes, and $a_{n}$ the number after $40(n-1)$ minutes. (Source: Hoppensteadt, F. and C. Peskin, Mathematics in Medicine and the Life Sciences, Springer-Verlag. ) (a) Write a formula for the $n$ th term $a_{n}$ of the geometric sequence $a_{1}, a_{2}, a_{3}, \ldots, a_{n}, \ldots$ (b) Determine the first value for $n$ where $a_{n}>1,000,000$ if $a_{1}=100$ (c) How long does it take for the number of bacteria to exceed 1 million?

Solve each problem .
(Modeling) Bacterial Growth The strain of bacteria described in Exercise 85 in the first section of this chapter will double in size and then divide every 40 minutes. Let $a_{1}$ be the initial number of bacteria cells, $a_{2}$ the number after 40 minutes, and $a_{n}$ the number after $40(n-1)$ minutes. (Source: Hoppensteadt, F. and C. Peskin, Mathematics in Medicine and the Life Sciences, Springer-Verlag. )
(a) Write a formula for the $n$ th term $a_{n}$ of the geometric sequence $a_{1}, a_{2}, a_{3}, \ldots, a_{n}, \ldots$
(b) Determine the first value for $n$ where $a_{n}>1,000,000$ if $a_{1}=100$
(c) How long does it take for the number of bacteria to exceed 1 million?

Key Concepts

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant ratio. This key concept is used to describe situations where processes change by a uniform percentage or factor over equal time intervals, making it essential in modeling phenomena such as bacterial growth.

Exponential Growth

Exponential growth occurs when the growth rate of a mathematical function is proportional to the current value, leading to the quantity increasing at an ever-accelerating rate. This concept is fundamental in understanding how populations, investments, or bacteria can grow rapidly over time by consistently multiplying by a constant factor.

Logarithmic Equations and Inequalities

Logarithms are the inverse operations of exponential functions and are crucial for solving equations where the variable is in the exponent. In the context of determining the time or step at which a growing process exceeds a certain threshold, logarithmic techniques allow one to isolate the exponent in an inequality or equation.

Time Conversion

Time conversion involves translating the number of periods or intervals into an actual time duration, and it is essential when dealing with models where each term corresponds to a fixed time period. This concept helps to contextualize the mathematical solution in real-world units such as minutes or hours.

Please give Ace some feedback