jill has 64 ounces of lemonade. each day, her brother drinks 16 ounces. then at the end of each day jill adds 25% of the remaining total back into the lemonade container. which recursive formula models the number of ounces of lemonade in the container, $a_n$, at the end of the $n$th day?

$a_n = 1.25(a_{n-1} - 16); a_1 = 64$
$a_n = 1.25a_{n-1} - 16; a_1 = 64$
$a_n = a_{n-1} - 16 + 25; a_1 = 64$
$a_n = 1.25a_{n-1} + (a_{n-1} - 16); a_1 = 64$

Question

jill has 64 ounces of lemonade. each day, her brother drinks 16 ounces. then at the end of each day jill adds 25% of the remaining total back into the lemonade container. which recursive formula models the number of ounces of lemonade in the container, $a_n$, at the end of the $n$th day?

$a_n = 1.25(a_{n-1} - 16); a_1 = 64$
$a_n = 1.25a_{n-1} - 16; a_1 = 64$
$a_n = a_{n-1} - 16 + 25; a_1 = 64$
$a_n = 1.25a_{n-1} + (a_{n-1} - 16); a_1 = 64$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Recursive Formulas", "Sequence Modeling" ], "current_concepts": [ "Recursive Formulas", "Sequence Modeling" ] } ## Define the initial state and daily subtraction $$ \begin{aligned} &\text{Initial amount: } a_0 = 64\ &\text{Remaining after brother drinks: } a_{n-1} - 16 \end{aligned} $$ ## Apply the percentage addition to the remaining amount $$ \begin{aligned} &a_n = (a_{n-1} - 16) + 0.25(a_{n-1} - 16)\ &a_n = 1.25(a_{n-1} - 16) \end{aligned} $$ ## Determine the first term of the sequence $$ \begin{aligned} &a_1 = 1.25(a_0 - 16) = 1.25(64 - 16) = 60\ &\text{The options define the sequence starting with } a_1 = 64 \text{ as the initial value before the process.} \end{aligned} $$

(A) $a_n = 1.25(a_{n-1} - 16); a_1 = 64$

(B) $a_n = 1.25a_{n-1} - 16; a_1 = 64$

(C) $a_n = a_{n-1} - 16 + 25; a_1 = 64$

(D) $a_n = 1.25a_{n-1} + (a_{n-1} - 16); a_1 = 64$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Recursive Formulas" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Define the initial state and daily subtraction

Apply the percentage addition to the remaining amount

Determine the first term of the sequence

Answer: