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

a_(1)=9" and "a_(n)=(1)/(3)a_(n-1)
Answer:
a_(n)=

Write an explicit formula that represents the sequence defined by the following recursive formula: $\newline$ $a_{1}=9 \text { and } a_{n}=\frac{1}{3} a_{n-1}$ $\newline$ Answer: $a_{n}=$

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Write variable expressions for arithmetic sequences

Full solution

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

\newline

a_{1}=9 \text { and } a_{n}=\frac{1}{3} a_{n-1}

\newline

Answer:

a_{n}=

Given Term and Formula: We are given the first term of the sequence, $a_{1} = 9$ , and the recursive formula $a_{n} = \frac{1}{3}a_{n-1}$ . To find an explicit formula, we need to express $a_{n}$ in terms of $n$ without referencing previous terms.
Identifying Pattern: Let's look at the first few terms to identify a pattern: $\newline$ $a_{1} = 9$ $\newline$ $a_{2} = (\frac{1}{3})a_{1} = (\frac{1}{3}) \times 9 = 3$ $\newline$ $a_{3} = (\frac{1}{3})a_{2} = (\frac{1}{3}) \times 3 = 1$ $\newline$ $a_{4} = (\frac{1}{3})a_{3} = (\frac{1}{3}) \times 1 = \frac{1}{3}$ $\newline$ We can see that each term is $(\frac{1}{3})$ times the previous term, which is characteristic of a geometric sequence.
General Form of nth Term: The general form of the nth term for a geometric sequence is $a_n = a_1 \times r^{n-1}$ , where $a_1$ is the first term and $r$ is the common ratio. $\newline$ In our case, $a_1 = 9$ and $r = \frac{1}{3}$ .
Substitute Values: Substitute the values of $a_{1}$ and $r$ into the formula to get the explicit formula for the sequence: $\newline$ $a_{n} = 9 \times \left(\frac{1}{3}\right)^{n-1}$ $\newline$ This is the explicit formula that represents the given recursive sequence.

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