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Write an explicit formula that represents the sequence defined by the following recursive formula:

a_(1)=75" and "a_(n)=(1)/(5)a_(n-1)
Answer:
a_(n)=

Write an explicit formula that represents the sequence defined by the following recursive formula: $\newline$ $a_{1}=75 \text { and } a_{n}=\frac{1}{5} a_{n-1}$ $\newline$ Answer: $a_{n}=$

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Evaluate an exponential function

Full solution

Q. Write an explicit formula that represents the sequence defined by the following recursive formula:

\newline

a_{1}=75 \text { and } a_{n}=\frac{1}{5} a_{n-1}

\newline

Answer:

a_{n}=

Identify Pattern: The recursive formula given is $a_n = \frac{1}{5}a_{n-1}$ , with the initial condition $a_{1} = 75$ . To find an explicit formula, we need to express $a_n$ in terms of $n$ without referencing previous terms.
Calculate Terms: Let's look at the first few terms to identify a pattern. We start with $a_{1} = 75$ . Using the recursive formula, $a_{2} = \frac{1}{5}a_{1} = \frac{1}{5} \times 75$ . Then, $a_{3} = \frac{1}{5}a_{2} = \frac{1}{5} \times \frac{1}{5} \times 75$ , and so on.
Derive Explicit Formula: We can see that each term is $(1/5)$ times the previous term. So, $a_{n}$ can be written as $a_{1} \times (1/5)^{n-1}$ , because each step we multiply by $(1/5)$ one more time than the previous step.
Substitute Initial Condition: Substituting the initial condition $a_{1} = 75$ into the pattern we found, the explicit formula becomes $a_{n} = 75 \times \left(\frac{1}{5}\right)^{n-1}.$

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