Question

How are exponential functions related to geometric sequence?

Answer 1

Hint: In this problem, we have to relate, the geometric sequence and the exponential function. Geometric sequences are formed by choosing a starting value and generating each subsequent value by multiplying the previous value by some constant called the geometric ratio. In an exponential function the inputs can be any real number from negative infinity to the positive infinity. Both the exponent and the geometric sequence have variables, that are exponent, an initial value and a constant ratio which is the base.

Complete step by step answer:
We should know that, Both the exponent and the geometric sequence have variables, that are exponent, an initial value and a constant ratio which is the base.
Exponential function is defined for all real numbers and geometric sequences are defined only for positive integers. Another difference is that the base of a geometric sequence can be negative but the base of an exponential function must be positive.
We can see an example,
For geometric sequence, the expression is
\[y={{x}^{2}}\]
For x = 10, y = 100.
For exponential function, the expression is
\[y={{2}^{x}}\]
For x = 10, y = 1024.
Here, exponential is much faster.

Note:
Students may get confused when relating the geometric sequence and exponential function. We should remember that a geometric sequence is a discrete version of exponential functions, which are continuous. We should know that, Both the exponent and the geometric sequence have variables, that are exponent, an initial value and a constant ratio which is the base.