Question

1- What is the connection between exponential function and geometric sequence.

Answer 1

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:

• a_n = a_1 * r^(n-1)

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:

• f(x) = a * b^x

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