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

a_(1)=8" and "a_(n)=a_(n-1)-2
Answer:
a_(n)=

Write an explicit formula that represents the sequence defined by the following recursive formula: $\newline$ $a_{1}=8 \text { and } a_{n}=a_{n-1}-2$ $\newline$ Answer: $a_{n}=$

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Convert an explicit formula to a recursive formula

Full solution

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

\newline

a_{1}=8 \text { and } a_{n}=a_{n-1}-2

\newline

Answer:

a_{n}=

Identify Pattern: To find the explicit formula for the given recursive sequence, we need to determine a pattern that relates the term number $n$ directly to its value without relying on the previous term.
Write Out Terms: Let's start by writing out the first few terms of the sequence using the recursive formula: $\newline$ $a_{1} = 8$ (given) $\newline$ $a_{2} = a_{1} - 2 = 8 - 2 = 6$ $\newline$ $a_{3} = a_{2} - 2 = 6 - 2 = 4$ $\newline$ $a_{4} = a_{3} - 2 = 4 - 2 = 2$ $\newline$ From this pattern, we can see that each term is $2$ less than the previous term.
Determine Relationship: Now, let's find the relationship between the term number $n$ and the term value. We can see that: $\newline$ $a_{1} = 8 = 8 - 2(1 - 1)$ $\newline$ $a_{2} = 6 = 8 - 2(2 - 1)$ $\newline$ $a_{3} = 4 = 8 - 2(3 - 1)$ $\newline$ $a_{4} = 2 = 8 - 2(4 - 1)$ $\newline$ It seems that the term value is equal to $8$ minus $2$ times $(n - 1)$ .
Generalize Formula: To confirm the pattern, let's generalize the formula: $\newline$ $a_{n} = 8 - 2(n - 1)$ $\newline$ Simplifying the formula, we get: $\newline$ $a_{n} = 8 - 2n + 2$ $\newline$ $a_{n} = 10 - 2n$ $\newline$ This is the explicit formula for the given sequence.

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