Exponential functions and geometric sequences are closely related, as the terms in a geometric sequence can be expressed using an exponential function.
A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant called the common ratio. For example, a geometric sequence starting with 2 and with a common ratio of 3 would be: 2, 6, 18, 54, 162, ...
The general formula for a geometric sequence is:
Where:
- a_n is the nth term of the sequence.
- a_1 is the first term of the sequence.
- r is the common ratio.
- n is the number of terms.
Exponential functions have the general form:
Where:
- a is a constant.
- b is the base.
- x is the exponent.
If we take the formula for a geometric sequence and rewrite it as an exponential function, we get:
- a_n = a_1 * r^(n-1) = a_1 * (1 + (r-1))^(n-1)
Using the binomial expansion formula, we can simplify this to:
- a_n = a_1 * (1 + (r-1))^(n-1) = a_1 * (1 + (r-1)*(n-1))
This is an exponential function with base (1 + (r-1)) and exponent (n-1). Therefore, the terms of a geometric sequence can be expressed using an exponential function.
Conversely, if we have an exponential function of the form f(x) = a * b^x, then we can find a geometric sequence by evaluating the function for a range of values of x. For example, if we take f(x) = 2 * 3^x and evaluate it for x = 1, 2, 3, 4, we get the geometric sequence 6, 18, 54, 162.
~ Zeph