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Answer :
We are given the sequence
[tex]$$
84,\; 56,\; \frac{112}{3},\; \frac{224}{9},\; \frac{448}{27},\; \frac{896}{81},\; \dots
$$[/tex]
and we want to determine a recursive formula for the sequence when [tex]$n>1$[/tex].
Step 1. Identify the pattern between consecutive terms
First, compute the ratio between the second and the first term:
[tex]$$
\frac{56}{84} = \frac{2}{3}.
$$[/tex]
Now, check the ratio between the third and the second term:
[tex]$$
\frac{\frac{112}{3}}{56} = \frac{112}{3} \cdot \frac{1}{56} = \frac{112}{168} = \frac{2}{3}.
$$[/tex]
Since the ratio between consecutive terms is constant and equal to [tex]$\frac{2}{3}$[/tex], the sequence is a geometric sequence.
Step 2. Write the recursive formula
For a geometric sequence with a common ratio [tex]$r = \frac{2}{3}$[/tex], the recursive formula is given by
[tex]$$
f(n) = \frac{2}{3} \, f(n-1) \quad \text{for } n > 1.
$$[/tex]
Thus, the recursive formula that defines this sequence for [tex]$n>1$[/tex] is
[tex]$$
\boxed{f(n) = \frac{2}{3} \, f(n-1)}.
$$[/tex]
This is the correct answer.
[tex]$$
84,\; 56,\; \frac{112}{3},\; \frac{224}{9},\; \frac{448}{27},\; \frac{896}{81},\; \dots
$$[/tex]
and we want to determine a recursive formula for the sequence when [tex]$n>1$[/tex].
Step 1. Identify the pattern between consecutive terms
First, compute the ratio between the second and the first term:
[tex]$$
\frac{56}{84} = \frac{2}{3}.
$$[/tex]
Now, check the ratio between the third and the second term:
[tex]$$
\frac{\frac{112}{3}}{56} = \frac{112}{3} \cdot \frac{1}{56} = \frac{112}{168} = \frac{2}{3}.
$$[/tex]
Since the ratio between consecutive terms is constant and equal to [tex]$\frac{2}{3}$[/tex], the sequence is a geometric sequence.
Step 2. Write the recursive formula
For a geometric sequence with a common ratio [tex]$r = \frac{2}{3}$[/tex], the recursive formula is given by
[tex]$$
f(n) = \frac{2}{3} \, f(n-1) \quad \text{for } n > 1.
$$[/tex]
Thus, the recursive formula that defines this sequence for [tex]$n>1$[/tex] is
[tex]$$
\boxed{f(n) = \frac{2}{3} \, f(n-1)}.
$$[/tex]
This is the correct answer.
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