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

a_(1)=1" and "a_(n)=6a_(n-1)
Answer:
a_(n)=

Write an explicit formula that represents the sequence defined by the following recursive formula: $\newline$ $a_{1}=1 \text { and } a_{n}=6 a_{n-1}$ $\newline$ Answer: $a_{n}=$

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Evaluate exponential functions

Full solution

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

\newline

a_{1}=1 \text { and } a_{n}=6 a_{n-1}

\newline

Answer:

a_{n}=

Identify Pattern: To find an explicit formula for the sequence, we start by looking at the first few terms to identify a pattern. $\newline$ Given $a_{1}=1$ , we can find the next few terms using the recursive formula $a_{n}=6a_{n-1}$ . $\newline$ $a_{2} = 6a_{1} = 6(1) = 6$ $\newline$ $a_{3} = 6a_{2} = 6(6) = 36$ $\newline$ $a_{4} = 6a_{3} = 6(36) = 216$ $\newline$ We can see that each term is $6$ times the previous term, which suggests that the sequence is geometric with a common ratio of $6$ .
Geometric Sequence Formula: The general form of an explicit formula for a geometric sequence is $a_n = a_1 \cdot r^{(n-1)}$ , where $a_1$ is the first term and $r$ is the common ratio. $\newline$ For our sequence, $a_1 = 1$ and $r = 6$ . Substituting these values into the formula gives us: $\newline$ $a_n = 1 \cdot 6^{(n-1)}$
Explicit Formula Simplification: Simplifying the formula, we get the explicit formula for the sequence: $\newline$ $a_{n} = 6^{(n-1)}$ $\newline$ This formula will give us the $n$ th term of the sequence for any positive integer $n$ .

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