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

a_(1)=4" and "a_(n)=a_(n-1)+5
Answer:
a_(n)=

Write an explicit formula that represents the sequence defined by the following recursive formula: $\newline$ $a_{1}=4 \text { and } a_{n}=a_{n-1}+5$ $\newline$ 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}=4 \text { and } a_{n}=a_{n-1}+5

\newline

Answer:

a_{n}=

Identify Pattern: To find the explicit formula for the sequence, we need to determine a pattern that can be applied directly to find any term in the sequence without having to find all the previous terms.
Find First Terms: Let's start by finding the first few terms of the sequence using the recursive formula: $\newline$ $a_{1} = 4$ (given) $\newline$ $a_{2} = a_{1} + 5 = 4 + 5 = 9$ $\newline$ $a_{3} = a_{2} + 5 = 9 + 5 = 14$ $\newline$ $a_{4} = a_{3} + 5 = 14 + 5 = 19$ $\newline$ From this pattern, we can see that each term is $5$ more than the previous term.
Analyze Relationship: Now, let's look at the relationship between the term number $n$ and the value of each term: $a_{1} = 4 = 4 + 5(1 - 1)$ $a_{2} = 9 = 4 + 5(2 - 1)$ $a_{3} = 14 = 4 + 5(3 - 1)$ $a_{4} = 19 = 4 + 5(4 - 1)$ We can see that the term value is equal to the first term $4$ plus $5$ times $(n - 1)$ .
Determine Explicit Formula: The pattern suggests that the explicit formula for the $n$ th term is: $\newline$ $a_{n} = 4 + 5(n - 1)$ $\newline$ This formula allows us to calculate the value of any term directly.
Verify Formula: To verify that this formula is correct, let's test it with $n = 5$ :
$a_{5} = 4 + 5(5 - 1) = 4 + 5(4) = 4 + 20 = 24$
Now, let's check it with the recursive definition:
$a_{5} = a_{4} + 5 = 19 + 5 = 24$
Both methods give us the same result, confirming that our explicit formula is correct.

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