To classify the given function [tex]\( f(x) = 10 \cdot 2^x \)[/tex], let's analyze its properties step-by-step:
1. Exponential Function:
The function [tex]\( f(x) = 10 \cdot 2^x \)[/tex] is of the form [tex]\( f(x) = a \cdot b^x \)[/tex], where [tex]\( a = 10 \)[/tex] and [tex]\( b = 2 \)[/tex]. This is a classic exponential function because it involves a constant base raised to the power of the variable [tex]\( x \)[/tex], scaled by a factor of 10.
2. Arithmetic Sequence:
An arithmetic sequence is defined by a common difference between consecutive terms and is given by the general form [tex]\( a_n = a + (n-1)d \)[/tex], where [tex]\( a \)[/tex] is the first term and [tex]\( d \)[/tex] is the common difference.
To check if our function is an arithmetic sequence, we would need it to satisfy the condition that there is a constant difference between consecutive outputs. For example, if we take two values [tex]\( f(n) \)[/tex] and [tex]\( f(n+1) \)[/tex]:
[tex]\[
f(n) = 10 \cdot 2^n
\][/tex]
[tex]\[
f(n+1) = 10 \cdot 2^{n+1} = 10 \cdot 2 \cdot 2^n = 20 \cdot 2^n
\][/tex]
The difference is:
[tex]\[
f(n+1) - f(n) = 20 \cdot 2^n - 10 \cdot 2^n = 10 \cdot 2^n (2 - 1) = 10 \cdot 2^n
\][/tex]
The difference itself is not constant since it depends on [tex]\( n \)[/tex]. Thus, [tex]\( f(x) \)[/tex] is not an arithmetic sequence.
3. Geometric Sequence:
A geometric sequence is defined by a common ratio between consecutive terms and is given by the general form [tex]\( a_n = a \cdot r^{n-1} \)[/tex], where [tex]\( a \)[/tex] is the first term and [tex]\( r \)[/tex] is the common ratio.
For our function to be a geometric sequence, we would need a constant ratio between consecutive outputs. Let's take two values [tex]\( f(n) \)[/tex] and [tex]\( f(n+1) \)[/tex]:
[tex]\[
f(n) = 10 \cdot 2^n
\][/tex]
[tex]\[
f(n+1) = 10 \cdot 2^{n+1} = 10 \cdot 2 \cdot 2^n = 20 \cdot 2^n
\][/tex]
The ratio is:
[tex]\[
\frac{f(n+1)}{f(n)} = \frac{20 \cdot 2^n}{10 \cdot 2^n} = 2
\][/tex]
While the ratio is constant, the definition of a geometric sequence typically implies discrete values of [tex]\( n \)[/tex], not continuous as with our exponent [tex]\( x \)[/tex] in [tex]\( 10 \cdot 2^x \)[/tex]. Hence, we distinguish this as an exponential function rather than a geometric sequence.
Therefore, by eliminating both arithmetic and geometric sequences, and recognizing the exponential nature of the function, we classify:
[tex]\[
\boxed{a \, function}
\][/tex]
