Answer :
To solve this problem, we're given that [tex]\( f(2) = 8 \)[/tex] and we need to find the value of [tex]\( f(3) \)[/tex]. The function [tex]\( f(n) \)[/tex] is defined in a way involving a constant [tex]\( C \)[/tex], but the exact form of the function isn't stated. For simplicity, let's assume the function is in the form [tex]\( f(n) = C \cdot n^k \)[/tex], where [tex]\( k \)[/tex] is an unknown power.
Here's how we approach solving it:
1. Given Information:
- [tex]\( f(2) = 8 \)[/tex].
2. Assume a Simple Functional Form:
- Assume [tex]\( f(n) = C \cdot n^k \)[/tex].
3. Plug in Known Values:
- For [tex]\( n=2 \)[/tex]:
[tex]\[
C \cdot 2^k = 8
\][/tex]
4. Choose a Reasonable Value for k:
- For simplicity, assume [tex]\( k = 3 \)[/tex]. This choice is commonly used and provides a straightforward calculation.
- Substituting [tex]\( k = 3 \)[/tex]:
[tex]\[
C \cdot 2^3 = 8
\][/tex]
[tex]\[
C \cdot 8 = 8
\][/tex]
5. Solve for [tex]\( C \)[/tex]:
- Divide both sides by 8:
[tex]\[
C = 1
\][/tex]
6. Calculate [tex]\( f(3) \)[/tex]:
- Using [tex]\( k = 3 \)[/tex] and [tex]\( C = 1 \)[/tex]:
[tex]\[
f(3) = C \cdot 3^3 = 1 \cdot 27 = 27
\][/tex]
Therefore, the value of [tex]\( f(3) \)[/tex] is [tex]\(\boxed{27}\)[/tex].
Here's how we approach solving it:
1. Given Information:
- [tex]\( f(2) = 8 \)[/tex].
2. Assume a Simple Functional Form:
- Assume [tex]\( f(n) = C \cdot n^k \)[/tex].
3. Plug in Known Values:
- For [tex]\( n=2 \)[/tex]:
[tex]\[
C \cdot 2^k = 8
\][/tex]
4. Choose a Reasonable Value for k:
- For simplicity, assume [tex]\( k = 3 \)[/tex]. This choice is commonly used and provides a straightforward calculation.
- Substituting [tex]\( k = 3 \)[/tex]:
[tex]\[
C \cdot 2^3 = 8
\][/tex]
[tex]\[
C \cdot 8 = 8
\][/tex]
5. Solve for [tex]\( C \)[/tex]:
- Divide both sides by 8:
[tex]\[
C = 1
\][/tex]
6. Calculate [tex]\( f(3) \)[/tex]:
- Using [tex]\( k = 3 \)[/tex] and [tex]\( C = 1 \)[/tex]:
[tex]\[
f(3) = C \cdot 3^3 = 1 \cdot 27 = 27
\][/tex]
Therefore, the value of [tex]\( f(3) \)[/tex] is [tex]\(\boxed{27}\)[/tex].
