$$ \begin{array}{l} ext { 40) Write a recursive formula for the explicit formula. } \ a_{n}=-7 n+52 \ a_{1}= \ \ a_{n}= \end{array} $$

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$$ \begin{array}{l}	ext { 40) Write a recursive formula for the explicit formula. } \ a_{n}=-7 n+52 \ a_{1}= \ \ a_{n}= \end{array} $$

UpStudy AI · Accepted Answer

We start with the explicit formula  
$$
a_n = -7n + 52.
$$

**Step 1. Find the first term:**  
Substitute $$ n = 1 $$:
$$
a_1 = -7(1) + 52 = -7 + 52 = 45.
$$

**Step 2. Compute the difference between consecutive terms:**  
The recursive formula is of the form  
$$
a_{n+1} = a_n + d.
$$  
To find the common difference $$ d $$, compute  
$$
a_{n+1} = -7(n+1) + 52 = -7n - 7 + 52 = -7n + 45.
$$  
Subtract $$ a_n $$:
$$
a_{n+1} - a_n = (-7n + 45) - (-7n + 52) = -7.
$$

Thus, the common difference is  
$$
d = -7.
$$

**Step 3. Write the recursive formula:**  
The recursive formula is  
$$
a_1 = 45,\quad a_{n+1} = a_n - 7.
$$
The recursive formula is: $$ a_1 = 45,\quad a_{n+1} = a_n - 7. $$

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