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

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

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

\newline

Answer:

a_{n}=

Identify Pattern: The recursive formula given is $a_{n}=a_{n-1}-5$ , with the initial condition $a_{1}=-7$ . To find the explicit formula, we need to determine the pattern of the sequence by looking at the first few terms.
Calculate First Few Terms: Let's calculate the first few terms using the recursive formula: $\newline$ $a_{2} = a_{1} - 5 = -7 - 5 = -12$ $\newline$ $a_{3} = a_{2} - 5 = -12 - 5 = -17$ $\newline$ $a_{4} = a_{3} - 5 = -17 - 5 = -22$ $\newline$ From these calculations, we can see that each term is $5$ less than the previous term, which suggests that the sequence is arithmetic with a common difference of $-5$ .
Use Arithmetic Sequence Formula: The $n$ th term of an arithmetic sequence can be found using the formula $a_n = a_1 + (n - 1)d$ , where $a_1$ is the first term and $d$ is the common difference. In this case, $a_1 = -7$ and $d = -5$ .
Substitute Values: Substituting the values of $a_{1}$ and $d$ into the formula, we get: $\newline$ $a_{n} = -7 + (n - 1)(-5)$ $\newline$ $a_{n} = -7 - 5n + 5$ $\newline$ $a_{n} = -5n - 2$ $\newline$ This is the explicit formula for the given recursive sequence.

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