the sixth term of an arithmetic sequence is $\frac{3}{2}$, and the twelfth term is $\frac{5}{2}$.

what is the common difference of the arithmetic sequence?

the common difference is .

Question

the sixth term of an arithmetic sequence is $\frac{3}{2}$, and the twelfth term is $\frac{5}{2}$.

what is the common difference of the arithmetic sequence?

the common difference is  .

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Arithmetic Sequences", "Arithmetic Sequence Explicit Formula" ], "new_concepts": [], "current_concepts": [ "Arithmetic Sequences", "Arithmetic Sequence Explicit Formula" ] } ## Set up the system of equations Using the **Arithmetic Sequence Explicit Formula** and **Arithmetic Sequences** knowledge points $$ \begin{aligned} a_n &= a_1 + (n-1)d \ a_6 &= a_1 + 5d = \frac{3}{2} \ a_{12} &= a_1 + 11d = \frac{5}{2} \end{aligned} $$ ## Solve for the common difference Using the **Arithmetic Sequences** knowledge point $$ \begin{aligned} (a_1 + 11d) - (a_1 + 5d) &= \frac{5}{2} - \frac{3}{2} \ 6d &= 1 \ d &= \frac{1}{6} \end{aligned} $$ The sixth term of an arithmetic sequence is $\frac{3}{2}$, and the twelfth term is $\frac{5}{2}$. What is the common difference of the arithmetic sequence? The common difference is $\frac{1}{6}$. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Arithmetic Sequences" ] }

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Question

Explanation:

Set up the system of equations

Solve for the common difference

Answer: