find the indicated arithmetic means for each set of nonconsecutive terms.

27. 3 means; 19 and -5
28. 5 means; -62 and -8
29. 4 means; 3 and 88
30. 8 means; -5.5 and 23.75
31. 7 means; -4.5 and 7.5
32. 10 means; 6 and 259

Question

find the indicated arithmetic means for each set of nonconsecutive terms.

27. 3 means; 19 and -5
28. 5 means; -62 and -8
29. 4 means; 3 and 88
30. 8 means; -5.5 and 23.75
31. 7 means; -4.5 and 7.5
32. 10 means; 6 and 259

Accepted Answer

{ "quality": "clear", "question_count": 6, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Arithmetic Sequences" ], "new_concepts": [ "Arithmetic Means" ], "current_concepts": [ "Arithmetic Sequences", "Arithmetic Means" ] } ## Calculate common differences for each sequence $$ \begin{aligned} &\text{27. } a_1 = 19, \ a_5 = -5 \implies d = \frac{-5 - 19}{5 - 1} = -6\ &\text{28. } a_1 = -62, \ a_7 = -8 \implies d = \frac{-8 - (-62)}{7 - 1} = 9\ &\text{29. } a_1 = 3, \ a_6 = 88 \implies d = \frac{88 - 3}{6 - 1} = 17\ &\text{30. } a_1 = -5.5, \ a_{10} = 23.75 \implies d = \frac{23.75 - (-5.5)}{10 - 1} = 3.25\ &\text{31. } a_1 = -4.5, \ a_9 = 7.5 \implies d = \frac{7.5 - (-4.5)}{9 - 1} = 1.5\ &\text{32. } a_1 = 6, \ a_{12} = 259 \implies d = \frac{259 - 6}{12 - 1} = 23 \end{aligned} $$ ## Generate the arithmetic means $$ \begin{aligned} &\text{27. } 19 + (-6) = 13, \ 13 + (-6) = 7, \ 7 + (-6) = 1\ &\text{28. } -62 + 9 = -53, \ -44, \ -35, \ -26, \ -17\ &\text{29. } 3 + 17 = 20, \ 37, \ 54, \ 71\ &\text{30. } -5.5 + 3.25 = -2.25, \ 1, \ 4.25, \ 7.5, \ 10.75, \ 14, \ 17.25, \ 20.5\ &\text{31. } -4.5 + 1.5 = -3, \ -1.5, \ 0, \ 1.5, \ 3, \ 4.5, \ 6\ &\text{32. } 6 + 23 = 29, \ 52, \ 75, \ 98, \ 121, \ 144, \ 167, \ 190, \ 213, \ 236 \end{aligned} $$ | No. | Answer | |---|---| | 27 | $13, 7, 1$ | | 28 | $-53, -44, -35, -26, -17$ | | 29 | $20, 37, 54, 71$ | | 30 | $-2.25, 1, 4.25, 7.5, 10.75, 14, 17.25, 20.5$ | | 31 | $-3, -1.5, 0, 1.5, 3, 4.5, 6$ | | 32 | $29, 52, 75, 98, 121, 144, 167, 190, 213, 236$ | { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Arithmetic Sequences" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate common differences for each sequence

Generate the arithmetic means

Answer:

No.	Answer
28	\(-53, -44, -35, -26, -17\)
29	\(20, 37, 54, 71\)
30	\(-2.25, 1, 4.25, 7.5, 10.75, 14, 17.25, 20.5\)
31	\(-3, -1.5, 0, 1.5, 3, 4.5, 6\)
32	\(29, 52, 75, 98, 121, 144, 167, 190, 213, 236\)