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$f(n)=-48*(-(1)/(4))^(n) Complete the recursive formula of f(n). {:[f(1)=◻],[f(n)=f(n-1)]:}$

$f(n)=-48 \cdot\left(-\frac{1}{4}\right)^{n}$ $\newline$ Complete the recursive formula of $f(n)$ . $\newline$ $\begin{array}{l} f(1)=\square \\ f(n)=f(n-1) \cdot \square \end{array}$

Full solution

f(n)=-48 \cdot\left(-\frac{1}{4}\right)^{n}

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Complete the recursive formula of

f(n)

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\begin{array}{l} f(1)=\square \\ f(n)=f(n-1) \cdot \square \end{array}

Given explicit formula: We are given the explicit formula for the sequence: $\newline$ $f(n) = -48 \cdot \left(-\left(\frac{1}{4}\right)\right)^n$ $\newline$ To find the recursive formula, we need to express $f(n)$ in terms of $f(n-1)$ . $\newline$ First, let's find $f(1)$ by substituting $n = 1$ into the explicit formula. $\newline$ $f(1) = -48 \cdot \left(-\left(\frac{1}{4}\right)\right)^1$ $\newline$ $f(1) = -48 \cdot \left(-\frac{1}{4}\right)$ $\newline$ $f(1) = \frac{48}{4}$ $\newline$ $f(1) = 12$
Find $f(1)$ : Now, let's find $f(2)$ to see the relationship between $f(2)$ and $f(1)$ .
$f(2) = -48 \times (-(1/4))^2$
$f(2) = -48 \times (1/16)$
$f(2) = -48/16$
$f(2) = -3$
Find $f(2)$ : We can see that to go from $f(1)$ to $f(2)$ , we multiply $f(1)$ by $-\left(\frac{1}{4}\right)$ :
$f(2) = f(1) \times -\left(\frac{1}{4}\right)$
$f(2) = 12 \times -\left(\frac{1}{4}\right)$
$f(2) = -3$
This relationship holds for each subsequent term in the sequence.
Relationship between $f(1)$ and $f(2)$ : Therefore, the recursive formula for the sequence is: $\newline$ $f(1) = 12$ $\newline$ $f(n) = f(n-1) \times -(\frac{1}{4})$ for n > 1