Connection Between Geometric Sequences and Exponential Functions
The connection between geometric sequences and exponential functions is a Grade 11 Algebra 1 topic from enVision Chapter 6 showing that a geometric sequence with first term a1 and common ratio r is the exponential function f(n) = a1 * r^(n-1). The sequence 3, 6, 12, 24 (ratio 2) becomes f(n) = 3 * 2^(n-1). The halving sequence 100, 50, 25 (ratio 0.5) becomes f(n) = 100 * (0.5)^(n-1). Geometric sequences are discrete versions of exponential functions — the domain is positive integers (term positions) rather than all real numbers, and the base b plays the role of the common ratio.
Key Concepts
A geometric sequence can be expressed as an exponential function. If $\{a n\}$ is a geometric sequence with first term $a 1$ and common ratio $r$, then: $$a n = a 1 \cdot r^{n 1}$$ This can be written as an exponential function $f(n) = a 1 \cdot r^{n 1}$ where the variable $n$ appears in the exponent.
Common Questions
How does a geometric sequence relate to an exponential function?
A geometric sequence IS an exponential function f(n) = a1 * r^(n-1), where r is the common ratio and the domain is positive integers.
Write the exponential function for the sequence 3, 6, 12, 24.
a1 = 3, r = 2. f(n) = 3 * 2^(n-1).
Write the exponential function for 100, 50, 25, 12.5.
a1 = 100, r = 0.5. f(n) = 100 * (0.5)^(n-1).
What is the domain of a geometric sequence as a function?
Positive integers {1, 2, 3, ...} representing the term positions. Unlike continuous exponential functions, sequences only have values at whole-number inputs.
Does a ratio < 1 mean the sequence is exponential decay?
Yes. If 0 < r < 1, terms decrease toward zero, corresponding to exponential decay in the equivalent function.
What role does n-1 play in f(n) = a1 * r^(n-1)?
It ensures f(1) = a1 (the first term). When n=1, r^(1-1) = r^0 = 1, so the function starts at a1.