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Exponential decay refers to the process where quantities decrease rapidly at a rate proportional to their current value. In our exercise, the sequence follows the function \( f(n) = 3 \times 2^{-n} \). Here, \( 2^{-n} \) is the exponent and represents the decay factor.
The negative exponent \( -n \) causes the base 2 to be less than 1 as \( n \) increases, making each term smaller. This creates what's known as exponential decay.
The constant 3 in the formula acts as an initial multiplier and affects the maximum value of the sequence, but as \( n \) becomes larger, the impact of \( 2^{-n} \) becomes dominant.

• For \( n = 0 \), \( f(0) = 3 \times 1 = 3 \)
• For \( n = 4 \), \( f(4) = 3 \times \frac{1}{16} = 0.1875 \)

This exponential decay explains why terms of the sequence decrease sharply with increasing \( n \).

Function evaluation involves substituting a specific value of \( n \) into a given function to find the corresponding output. For our sequence, we use the function \( f(n) = 3 \times 2^{-n} \) to determine each term.
Let's look at how the function evaluates for different \( n \) values:

• For \( n = 0 \), substitute into the function: \( f(0) = 3 \times 2^{0} = 3 \times 1 = 3 \)
• For \( n = 1 \), substitute into the function: \( f(1) = 3 \times 2^{-1} = 3 \times \frac{1}{2} = 1.5 \)
• Each subsequent term is found by increasing \( n \) and calculating accordingly.

The process of function evaluation helps us systematically find each term in a sequence, and understand how the formula produces these terms.

Sequence terms are the individual outputs when a sequence function is evaluated for consecutive positive integers. Our sequence \( f(n) = 3 \times 2^{-n} \) is infinite, but we're focused on the first five terms.
To determine sequence terms, evaluate the sequence function for initial values of \( n \):

• Term 1: \( n = 0 \rightarrow f(0) = 3 \)
• Term 2: \( n = 1 \rightarrow f(1) = 1.5 \)
• Term 3: \( n = 2 \rightarrow f(2) = 0.75 \)
• Term 4: \( n = 3 \rightarrow f(3) = 0.375 \)
• Term 5: \( n = 4 \rightarrow f(4) = 0.1875 \)

The terms begin with a larger value (3) and get progressively smaller as \( n \) increases, illustrating the concept of exponential decay.
Each term represents a point on the function derived from specific \( n \)-values. Understanding each term helps grasp how sequences represent mathematical concepts like decay or growth over successive integers.