Arithmetic and Geometric Sequences as Functions - Algebra 2

1. Model: A population starts at $800$ and grows by $6\%$ each year. What is $a_n$?

   $a_n=800\cdot 1.06^{n-1}$. Geometric sequence: each year multiplied by growth factor $1.06$.

2. What is the explicit formula for the arithmetic sequence with $a_1=12$ and $d=-3$?

   $a_n=12+(n-1)(-3)$. Uses the standard explicit formula with $a_1=12$ and $d=-3$.

3. Find the common difference $d$ if an arithmetic sequence has $a_3=10$ and $a_8=35$.

   $d=5$. Using $d=\frac{a_m-a_k}{m-k}$: $d=\frac{35-10}{8-3}=\frac{25}{5}=5$.

4. What is the explicit formula for the geometric sequence with $a_1=2$ and $r=\frac{1}{4}$?

   $a_n=2\left(\frac{1}{4}\right)^{n-1}$. Uses the standard explicit formula with $a_1=2$ and $r=\frac{1}{4}$.

5. Find $a_7$ for the arithmetic sequence $a_n=2+5(n-1)$.

   $a_7=32$. Using $a_n=2+5(n-1)$: $a_7=2+5(6)=2+30=32$.

6. Model: A bouncing ball reaches $120$ cm, then $75\%$ of the previous height each bounce. What is $a_n$?

   $a_n=120\cdot 0.75^{n-1}$. Geometric sequence: each bounce multiplied by factor $0.75$.

7. Model: A car value is $\$18{,}000$ and depreciates $12\%$ yearly. What is $a_n$?

   $a_n=18000\cdot 0.88^{n-1}$. Geometric sequence: each year multiplied by decay factor $0.88$.

8. Model: A salary starts at $\$40{,}000$ and increases by $\$1{,}500$ yearly. What is $a_n$?

   $a_n=40000+1500(n-1)$. Arithmetic sequence: constant yearly increase of $\$1{,}500$.

9. Translate to explicit form: $a_1=5$, $a_n=\frac{1}{2}a_{n-1}$ for $n\ge^2$.

   $a_n=5\left(\frac{1}{2}\right)^{n-1}$. Converting recursive to explicit: $r=\frac{1}{2}$, so $a_n=5\left(\frac{1}{2}\right)^{n-1}$.

10. Use $a_n=a_k\cdot r^{n-k}$: If $a_3=16$ and $r=\frac{1}{2}$, what is $a_7$?

    $a_7=1$. Using the general form: $a_7=16\cdot\left(\frac{1}{2}\right)^{7-3}=16\cdot\frac{1}{16}=1$.

11. What is the explicit formula for an arithmetic sequence with first term $a_1$ and common difference $d$?

    $a_n=a_1+(n-1)d$. Standard explicit form adds $(n-1)$ copies of $d$ to the first term.

12. Find the explicit formula if an arithmetic sequence has $a_1=-1$ and $a_5=11$.

    $a_n=-1+3(n-1)$. From $a_1=-1$ and $a_5=11$: $d=\frac{11-(-1)}{5-1}=3$.

13. Identify the missing term to keep it arithmetic: $2,\,\square,\,10,\,14$.

    $6$. Arithmetic mean of $2$ and $10$ gives the middle term.

14. Find $a_5$ for the geometric sequence with $a_1=3$ and $r=2$.

    $a_5=48$. Using $a_n=a_1\cdot r^{n-1}$: $a_5=3\cdot 2^{5-1}=3\cdot 16=48$.

15. What is the explicit formula for an arithmetic sequence written from term $a_k$ instead of $a_1$?

    $a_n=a_k+(n-k)d$. General explicit form starting from any known term $a_k$.

16. Find $r$ if a geometric sequence has $a_n=10\cdot\left(\frac{3}{2}\right)^{n-1}$.

    $r=\frac{3}{2}$. The base of the exponent gives the common ratio directly.

17. What is the explicit formula for a geometric sequence written from term $a_k$ instead of $a_1$?

    $a_n=a_k\cdot r^{n-k}$. General explicit form starting from any known term $a_k$.

18. Find the geometric mean between $4$ and $36$ (assume positive).

    $12$. Using $\sqrt{ab}$: $\sqrt{4\cdot 36}=\sqrt{144}=12$.

19. What is the formula for the common ratio $r$ using two terms $a_k$ and $a_m$ of a geometric sequence?

    $r=\left(\frac{a_m}{a_k}\right)^{\frac{1}{m-k}}$. Ratio formula using the $(m-k)$th root of the term quotient.

20. Identify whether $a_n=2n+1$ defines an arithmetic or geometric sequence.

    Arithmetic. Linear form $a_n=2n+1$ has constant differences between consecutive terms.

21. Find and correct the error: Arithmetic explicit written as $a_n=a_1+nd$.

    Correct: $a_n=a_1+(n-1)d$. The exponent should be $(n-1)$, not $n$, for standard form.

22. Find the recursive form if a geometric sequence has $a_1=10$ and $a_2=-5$.

    $a_1=10$; $a_n=-\frac{1}{2}a_{n-1}$. From consecutive terms: $r=\frac{a_2}{a_1}=\frac{-5}{10}=-\frac{1}{2}$.

23. Identify the arithmetic sequence: $3,7,11,15,\dots$; what is the common difference $d$?

    $d=4$. Each term increases by $4$: $7-3=4$, $11-7=4$, etc.

24. What is the recursive definition for the geometric sequence with $a_1=9$ and $r=-2$?

    $a_1=9$; $a_n=-2\,a_{n-1}$. Standard recursive form with given first term and ratio.

25. Translate to recursive form: $a_n=8\cdot(-3)^{n-1}$ with $n\ge^1$.

    $a_1=8$; $a_n=-3\,a_{n-1}$. Converting explicit to recursive: $a_1=8$ and $r=-3$.

26. Find $a_1$ for a geometric sequence with $a_3=36$ and $r=3$.

    $a_1=4$. Working backwards: $a_1=\frac{a_3}{r^2}=\frac{36}{3^2}=\frac{36}{9}=4$.

27. Translate to explicit form: $a_1=7$, $a_n=a_{n-1}-2$ for $n\ge^2$.

    $a_n=7+(n-1)(-2)$. Converting recursive to explicit: $d=-2$, so $a_n=7+(n-1)(-2)$.

28. Identify which is geometric: $a_n=12-3n$ or $a_n=12\left(-3\right)^{n-1}$?

    $a_n=12\left(-3\right)^{n-1}$. Exponential function has constant ratios; linear has constant differences.

29. Find $a_6$ for the arithmetic sequence with $a_1=4$ and $d=7$.

    $a_6=39$. Using $a_n=a_1+(n-1)d$: $a_6=4+(6-1)(7)=4+35=39$.

30. Identify which is arithmetic: $a_n=5n-2$ or $a_n=5\cdot 2^{n-1}$?

    $a_n=5n-2$. Linear function has constant differences; exponential has constant ratios.

31. Identify the geometric sequence: $5,15,45,135,\dots$; what is the common ratio $r$?

    $r=3$. Each term is multiplied by $3$: $15÷5=3$, $45÷15=3$, etc.

32. Find the arithmetic mean between $8$ and $20$.

    $14$. Using $\frac{a+b}{2}$: $\frac{8+20}{2}=\frac{28}{2}=14$.

33. What is the explicit formula for a geometric sequence with first term $a_1$ and common ratio $r$?

    $a_n=a_1\cdot r^{n-1}$. Standard explicit form multiplies $a_1$ by $r$ raised to power $(n-1)$.

34. What is the arithmetic mean formula that gives the middle term between $a$ and $b$ in an arithmetic sequence?

    $\frac{a+b}{2}$. Average of two consecutive terms in an arithmetic sequence.

35. Model: You save $50 the first week and increase by $10 weekly. What is $a_n$?

    $a_n=50+10(n-1)$. Arithmetic sequence: constant weekly increase of $10.

36. Identify whether $a_n=7\cdot 5^{n}$ defines an arithmetic or geometric sequence.

    Geometric. Exponential form $a_n=7\cdot 5^n$ has constant ratios between consecutive terms.

37. Find $a_6$ for the geometric sequence $a_n=\left(-2\right)\cdot 3^{n-1}$.

    $a_6=-486$. Using $a_n=(-2)\cdot 3^{n-1}$: $a_6=(-2)\cdot 3^5=(-2)\cdot 243=-486$.

38. What is the recursive definition for the arithmetic sequence with $a_1=-5$ and $d=6$?

    $a_1=-5$; $a_n=a_{n-1}+6$. Standard recursive form with given first term and difference.

39. What is the recursive definition for an arithmetic sequence with first term $a_1$ and difference $d$?

    $a_1$ given; $a_n=a_{n-1}+d$ for $n\ge^2$. Each term equals the previous term plus the common difference.

40. What is the formula for the common difference $d$ using two terms $a_k$ and $a_m$ of an arithmetic sequence?

    $d=\frac{a_m-a_k}{m-k}$. Difference formula: change in terms divided by change in positions.

41. Which option is the common ratio for the geometric sequence $-4,12,-36,108,\dots$?

    $r=-3$. Each term is multiplied by $-3$: $12÷(-4)=-3$.

42. Find and correct the error: Geometric explicit written as $a_n=a_1\cdot r^{n}$.

    Correct: $a_n=a_1\cdot r^{n-1}$. The exponent should be $(n-1)$, not $n$, for standard form.

43. Translate to recursive form: $a_n=3+4(n-1)$ with $n\ge^1$.

    $a_1=3$; $a_n=a_{n-1}+4$. Converting explicit to recursive: $a_1=3$ and $d=4$.

44. Find the common ratio $r$ if a geometric sequence has $a_2=6$ and $a_5=48$.

    $r=2$. Using $r=\left(\frac{a_m}{a_k}\right)^{\frac{1}{m-k}}$: $r=\left(\frac{48}{6}\right)^{\frac{1}{3}}=8^{\frac{1}{3}}=2$.

45. Identify the missing term to keep it geometric: $3,\,\square,\,12,\,24$.

    $6$. Geometric mean of $3$ and $12$ gives the middle term.

46. Which option is the common difference for the arithmetic sequence $-2,1,4,7,\dots$?

    $d=3$. Each term increases by the same amount: $1-(-2)=3$.

47. Find the recursive form if an arithmetic sequence has $a_1=9$ and $a_2=4$.

    $a_1=9$; $a_n=a_{n-1}-5$. From consecutive terms: $d=a_2-a_1=4-9=-5$.

48. What is the geometric mean formula that gives the middle positive term between $a$ and $b$ in a geometric sequence?

    $\sqrt{ab}$. Square root of the product of two consecutive positive terms.

49. Find $d$ if an arithmetic sequence has $a_n=18-6(n-1)$.

    $d=-6$. The coefficient of $(n-1)$ gives the common difference directly.

50. Use $a_n=a_k+(n-k)d$: If $a_4=9$ and $d=2$, what is $a_{10}$?

    $a_{10}=21$. Using the general form: $a_{10}=9+(10-4)(2)=9+12=21$.

51. Find the explicit formula if a geometric sequence has $a_1=6$ and $a_4=48$.

    $a_n=6\cdot 2^{n-1}$. From $a_1=6$ and $a_4=48$: $r=\left(\frac{48}{6}\right)^{\frac{1}{3}}=2$.

52. Find $a_1$ for an arithmetic sequence with $a_4=20$ and $d=3$.

    $a_1=11$. Working backwards: $a_1=a_4-3d=20-3(3)=20-9=11$.

53. What is the recursive definition for a geometric sequence with first term $a_1$ and ratio $r$?

    $a_1$ given; $a_n=r\cdot a_{n-1}$ for $n\ge^2$. Each term equals the previous term multiplied by the common ratio.

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