in an arithmetic sequence, $a_{14} = -75$ and $a_{26} = -123$. which recursive formula defines the sequence?

$a_n = a_{n-1} - 4; a_1 = -23$
$a_n = a_{n-1} - 4; a_1 = -19$
$a_n = -4a_{n-1} - 19; a_1 = -23$
$a_n = -4a_{n-1} - 19; a_1 = -19$

Question

in an arithmetic sequence, $a_{14} = -75$ and $a_{26} = -123$. which recursive formula defines the sequence?

$a_n = a_{n-1} - 4; a_1 = -23$
$a_n = a_{n-1} - 4; a_1 = -19$
$a_n = -4a_{n-1} - 19; a_1 = -23$
$a_n = -4a_{n-1} - 19; a_1 = -19$

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": [ "Arithmetic Sequences" ], "current_concepts": [ "Arithmetic Sequences", "Recursive Formulas" ] } ## Find the common difference $$ \begin{aligned} d &= \frac{a_{26} - a_{14}}{26 - 14} \ d &= \frac{-123 - (-75)}{12} \ d &= \frac{-48}{12} = -4 \end{aligned} $$ ## Find the first term $$ \begin{aligned} a_{14} &= a_1 + 13d \ -75 &= a_1 + 13(-4) \ -75 &= a_1 - 52 \ a_1 &= -23 \end{aligned} $$ ## Write the recursive formula $$ \begin{aligned} a_n &= a_{n-1} - 4 \ a_1 &= -23 \end{aligned} $$

(A) $a_n = a_{n-1} - 4; a_1 = -23$

(B) $a_n = a_{n-1} - 4; a_1 = -19$

(C) $a_n = -4a_{n-1} - 19; a_1 = -23$

(D) $a_n = -4a_{n-1} - 19; a_1 = -19$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Arithmetic Sequences" ] }

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QUESTION IMAGE

Question

Explanation:

Find the common difference

Find the first term

Write the recursive formula

Answer: