Let p(t) represent the consumer price index. If the rate of...

Let $p(t)$ represent the consumer price index. If the rate of inflation of the price index is constant at $5 \%$ (i.e., the growth rate of $p(t)$ is $5 \%$ ), and the price index has a base value of 100 at time $t=0$, solve for the expression showing the price index as a function of time.

Key Concepts

Exponential Growth

Exponential growth describes a process in which the rate of increase is directly proportional to the current value, leading to continuous and compounded increases over time. This concept is essential in scenarios where quantities grow by a fixed percentage per time unit, such as in modeling economic variables or populations.

Constant Relative Growth Rate

A constant relative growth rate signifies that the percentage change per unit time remains steady. This assumption simplifies the mathematical modeling of growth processes by yielding an exponential function, where the proportionality factor remains unaltered, and is widely applicable in finance, economics, and natural sciences.

Initial Value Problem

An initial value problem involves solving a differential equation given a specific starting condition. The initial condition provides the necessary anchor point to determine the unique solution of the model over time, ensuring that the mathematical representation aligns with the observed or prescribed starting state.

Differential Equations

Differential equations are mathematical equations that relate a function to its derivatives, representing rates of change. They are fundamental in describing dynamic systems, where dependencies on time or other variables are modeled to understand how the system evolves from an initial state under specific conditions.

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