Answer :
To solve the problem, we need to find [tex]\( f(1) \)[/tex] given the recursive relationship for the sequence:
[tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], with the additional information that [tex]\( f(3) = 9 \)[/tex].
Let's work through it step-by-step:
1. Find [tex]\( f(2) \)[/tex]:
- We know that [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex].
- Since [tex]\( f(3) = 9 \)[/tex], we can set up the equation:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
[tex]\[
9 = \frac{1}{3} f(2)
\][/tex]
- To find [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]
2. Find [tex]\( f(1) \)[/tex]:
- Similarly, from the recursive relationship, [tex]\( f(2) = \frac{1}{3} f(1) \)[/tex].
- We found that [tex]\( f(2) = 27 \)[/tex], so:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
[tex]\[
27 = \frac{1}{3} f(1)
\][/tex]
- To find [tex]\( f(1) \)[/tex], multiply both sides by 3:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
[tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], with the additional information that [tex]\( f(3) = 9 \)[/tex].
Let's work through it step-by-step:
1. Find [tex]\( f(2) \)[/tex]:
- We know that [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex].
- Since [tex]\( f(3) = 9 \)[/tex], we can set up the equation:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
[tex]\[
9 = \frac{1}{3} f(2)
\][/tex]
- To find [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]
2. Find [tex]\( f(1) \)[/tex]:
- Similarly, from the recursive relationship, [tex]\( f(2) = \frac{1}{3} f(1) \)[/tex].
- We found that [tex]\( f(2) = 27 \)[/tex], so:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
[tex]\[
27 = \frac{1}{3} f(1)
\][/tex]
- To find [tex]\( f(1) \)[/tex], multiply both sides by 3:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
