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$f(n)=45\cdot\left(\dfrac{4}{5}\right)^{\large{\,n-1}}$ Complete the recursive formula of $f(n)$

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Simplify radical expressions using conjugates

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Q.

f(n)=45\cdot\left(\dfrac{4}{5}\right)^{\large{\,n-1}}

Complete the recursive formula of

f(n)

Identify Initial Term: Identify the initial term of the sequence. $\newline$ The initial term $f(1)$ is when $n=1$ . $\newline$ $f(1) = 45 \times \left(\frac{4}{5}\right)^{1-1}$ $\newline$ $f(1) = 45 \times \left(\frac{4}{5}\right)^0$ $\newline$ $f(1) = 45 \times 1$ $\newline$ $f(1) = 45$
Calculate Second Term: Calculate the second term to find the common ratio. $\newline$ The second term $f(2)$ is when $n=2$ . $\newline$ $f(2) = 45 \times (\frac{4}{5})^{2-1}$ $\newline$ $f(2) = 45 \times (\frac{4}{5})^1$ $\newline$ $f(2) = 45 \times (\frac{4}{5})$ $\newline$ $f(2) = 36$
Determine Common Ratio: Determine the common ratio by dividing the second term by the first term. $\newline$ The common ratio $r$ is $f(2) / f(1)$ . $\newline$ $r = 36 / 45$ $\newline$ $r = 4/5$
Write Recursive Formula: Write the recursive formula using the initial term and the common ratio. $\newline$ The recursive formula for a geometric sequence is $f(n) = f(n-1) \times r$ , where $r$ is the common ratio. $\newline$ For this sequence, the recursive formula is: $\newline$ $f(n) = f(n-1) \times \left(\frac{4}{5}\right)$ , for $n > 1$ $\newline$ And the initial term is $f(1) = 45$ .

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