find both an explicit formula and a recursive formula for the $n$th term of each arithmetic sequence.
10. $2, 5, 8, \dots$
11. $-6, 5, 16, \dots$
12. $-9, -16, -23, \dots$
13. $4, 19, 34, \dots$
14. $25, 11, -3, \dots$
15. $7, -3.5, -14, \dots$
16. $-18, 4, 26, \dots$
17. $1, 37, 73, \dots$

find the specified value for the arithmetic sequence with the given characteristics.
18. if $a_{14} = 85$ and $d = 9$, find $a_1$.
19. find $d$ for $-24, -31, -38, \dots$
20. if $a_n = 14$, $a_1 = -36$, and $d = 5$, find $n$.
21. if $a_1 = 47$ and $d = -5$, find $a_{12}$.
22. if $a_{22} = 95$ and $a_1 = 11$, find $d$.
23. find $a_6$ for $84, 5, -74, \dots$
24. if $a_n = -20$, $a_1 = 46$, and $d = -11$, find $n$.
25. if $a_{35} = -63$ and $a_1 = 39$, find $d$.

Question

find both an explicit formula and a recursive formula for the $n$th term of each arithmetic sequence.
10. $2, 5, 8, \dots$
11. $-6, 5, 16, \dots$
12. $-9, -16, -23, \dots$
13. $4, 19, 34, \dots$
14. $25, 11, -3, \dots$
15. $7, -3.5, -14, \dots$
16. $-18, 4, 26, \dots$
17. $1, 37, 73, \dots$

find the specified value for the arithmetic sequence with the given characteristics.
18. if $a_{14} = 85$ and $d = 9$, find $a_1$.
19. find $d$ for $-24, -31, -38, \dots$
20. if $a_n = 14$, $a_1 = -36$, and $d = 5$, find $n$.
21. if $a_1 = 47$ and $d = -5$, find $a_{12}$.
22. if $a_{22} = 95$ and $a_1 = 11$, find $d$.
23. find $a_6$ for $84, 5, -74, \dots$
24. if $a_n = -20$, $a_1 = 46$, and $d = -11$, find $n$.
25. if $a_{35} = -63$ and $a_1 = 39$, find $d$.

Accepted Answer

{ "quality": "clear", "question_count": 16, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Arithmetic Sequences", "Explicit Formula", "Recursive Formula" ], "new_concepts": [], "current_concepts": [ "Arithmetic Sequences", "Explicit Formula", "Recursive Formula" ] } ## Identify sequences and characteristics for even-numbered questions 10 to 16 $$ \begin{aligned} &\text{10. } 2, 5, 8, \dots \implies a_1 = 2, \quad d = 3\ &\text{12. } -9, -16, -23, \dots \implies a_1 = -9, \quad d = -7\ &\text{14. } 25, 11, -3, \dots \implies a_1 = 25, \quad d = -14\ &\text{16. } -18, 4, 26, \dots \implies a_1 = -18, \quad d = 22 \end{aligned} $$ ## Derive explicit and recursive formulas for questions 10 to 16 $$ \begin{aligned} &\text{10. } a_n = 2 + (n-1)(3) = 3n - 1; \quad a_1 = 2, \ a_n = a_{n-1} + 3\ &\text{12. } a_n = -9 + (n-1)(-7) = -7n - 2; \quad a_1 = -9, \ a_n = a_{n-1} - 7\ &\text{14. } a_n = 25 + (n-1)(-14) = -14n + 39; \quad a_1 = 25, \ a_n = a_{n-1} - 14\ &\text{16. } a_n = -18 + (n-1)(22) = 22n - 40; \quad a_1 = -18, \ a_n = a_{n-1} + 22 \end{aligned} $$ ## Solve even-numbered characteristic questions 18 to 24 $$ \begin{aligned} &\text{18. } a_{14} = a_1 + 13d \implies 85 = a_1 + 13(9) \implies a_1 = 85 - 117 = -32\ &\text{20. } a_n = a_1 + (n-1)d \implies 14 = -36 + (n-1)5 \implies 50 = 5(n-1) \implies n = 11\ &\text{22. } a_{22} = a_1 + 21d \implies 95 = 11 + 21d \implies 84 = 21d \implies d = 4\ &\text{24. } a_n = a_1 + (n-1)d \implies -20 = 46 + (n-1)(-11) \implies -66 = -11(n-1) \implies n = 7 \end{aligned} $$ | No. | Problem | Answer | |---|---|---| | 10 | Find explicit and recursive formulas for $2, 5, 8, \dots$ | Explicit: $a_n = 3n - 1$
Recursive: $a_1 = 2, \ a_n = a_{n-1} + 3$ | | 12 | Find explicit and recursive formulas for $-9, -16, -23, \dots$ | Explicit: $a_n = -7n - 2$
Recursive: $a_1 = -9, \ a_n = a_{n-1} - 7$ | | 14 | Find explicit and recursive formulas for $25, 11, -3, \dots$ | Explicit: $a_n = -14n + 39$
Recursive: $a_1 = 25, \ a_n = a_{n-1} - 14$ | | 16 | Find explicit and recursive formulas for $-18, 4, 26, \dots$ | Explicit: $a_n = 22n - 40$
Recursive: $a_1 = -18, \ a_n = a_{n-1} + 22$ | | 18 | If $a_{14} = 85$ and $d = 9$, find $a_1$. | $a_1 = -32$ | | 20 | If $a_n = 14$, $a_1 = -36$, and $d = 5$, find $n$. | $n = 11$ | | 22 | If $a_{22} = 95$ and $a_1 = 11$, find $d$. | $d = 4$ | | 24 | If $a_n = -20$, $a_1 = 46$, and $d = -11$, find $n$. | $n = 7$ | { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Arithmetic Sequences" ] }

No.	Problem	Answer
12	Find explicit and recursive formulas for \(-9, -16, -23, \dots\)	Explicit: \(a_n = -7n - 2\)<br>Recursive: \(a_1 = -9, \ a_n = a_{n-1} - 7\)
14	Find explicit and recursive formulas for \(25, 11, -3, \dots\)	Explicit: \(a_n = -14n + 39\)<br>Recursive: \(a_1 = 25, \ a_n = a_{n-1} - 14\)
16	Find explicit and recursive formulas for \(-18, 4, 26, \dots\)	Explicit: \(a_n = 22n - 40\)<br>Recursive: \(a_1 = -18, \ a_n = a_{n-1} + 22\)
18	If \(a_{14} = 85\) and \(d = 9\), find \(a_1\).	\(a_1 = -32\)
20	If \(a_n = 14\), \(a_1 = -36\), and \(d = 5\), find \(n\).	\(n = 11\)
22	If \(a_{22} = 95\) and \(a_1 = 11\), find \(d\).	\(d = 4\)
24	If \(a_n = -20\), \(a_1 = 46\), and \(d = -11\), find \(n\).	\(n = 7\)

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QUESTION IMAGE

Question

Explanation:

Identify sequences and characteristics for even-numbered questions 10 to 16

Derive explicit and recursive formulas for questions 10 to 16

Solve even-numbered characteristic questions 18 to 24

Answer: