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

a_(1)=7" and "a_(n)=5a_(n-1)
Answer:
a_(n)=

Write an explicit formula that represents the sequence defined by the following recursive formula: $\newline$ $a_{1}=7 \text { and } a_{n}=5 a_{n-1}$ $\newline$ Answer: $a_{n}=$

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Convert between explicit and recursive formulas

Full solution

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

\newline

a_{1}=7 \text { and } a_{n}=5 a_{n-1}

\newline

Answer:

a_{n}=

Identify Base Case: Identify the base case and the recursive step from the given recursive formula. $\newline$ The base case is $a_{1} = 7$ , which means the first term of the sequence is $7$ . The recursive step is $a_{n} = 5a_{n-1}$ , which means each term is $5$ times the previous term. This indicates that the sequence is geometric with a common ratio of $5$ .
Write First Few Terms: Write the first few terms of the sequence using the recursive formula to observe the pattern. $\newline$ $a_{1} = 7$ $\newline$ $a_{2} = 5a_{1} = 5 \times 7 = 35$ $\newline$ $a_{3} = 5a_{2} = 5 \times 35 = 175$ $\newline$ $a_{4} = 5a_{3} = 5 \times 175 = 875$ $\newline$ We can see that each term is $5$ times the previous term, confirming that the sequence is geometric.
Derive Explicit Formula: Derive the explicit formula for the geometric sequence. $\newline$ For a geometric sequence, the nth term is given by $a_n = a_1 \cdot r^{n-1}$ , where $a_1$ is the first term and $r$ is the common ratio. In this case, $a_1 = 7$ and $r = 5$ . Therefore, the explicit formula is $a_n = 7 \cdot 5^{n-1}$ .

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