each year, a daylily farm sells a portion of their daylilies and allows a portion to grow and divide. the recursive formula $a_n = 1.5(a_{n-1}) - 100$ represents the number of daylilies, $a$, on the farm after $n$ years. after the fifth year, the farmers estimate they have 2,225 daylilies. how many daylilies were on the farm after the first year?

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Question

each year, a daylily farm sells a portion of their daylilies and allows a portion to grow and divide. the recursive formula \$a_n = 1.5(a_{n-1}) - 100\$ represents the number of daylilies, \$a\$, on the farm after \$n\$ years. after the fifth year, the farmers estimate they have 2,225 daylilies. how many daylilies were on the farm after the first year?

279
518
600
800

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Recursive Formulas" ], "new_concepts": [], "current_concepts": [ "Recursive Formulas", "Sequence Modeling" ] } ## Express the recursive relation backward $$ \begin{aligned} a_n &= 1.5 a_{n-1} - 100 \ 1.5 a_{n-1} &= a_n + 100 \ a_{n-1} &= \frac{a_n + 100}{1.5} \end{aligned} $$ ## Calculate preceding terms from fifth year $$ \begin{aligned} a_5 &= 2225 \ a_4 &= \frac{2225 + 100}{1.5} = \frac{2325}{1.5} = 1550 \ a_3 &= \frac{1550 + 100}{1.5} = \frac{1650}{1.5} = 1100 \ a_2 &= \frac{1100 + 100}{1.5} = \frac{1200}{1.5} = 800 \ a_1 &= \frac{800 + 100}{1.5} = \frac{900}{1.5} = 600 \end{aligned} $$

279

518

600

800

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

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QUESTION IMAGE

Question

Explanation:

Express the recursive relation backward

Calculate preceding terms from fifth year

Answer: