Arithmetic and Geometric Sequences as Functions - Algebra 1

1. Model: A car value is $\$20{,}000$ and keeps $0.85$ of its value yearly. Write $V_n$ explicitly.

   $V_n = 20000\left(0.85\right)^{n-1}$. Geometric sequence models exponential decay with factor $0.85$.

2. Find $a_6$ for the geometric sequence $a_1 = 1$ and $r = \frac{1}{3}$.

   $a_6 = \frac{1}{243}$. Use $a_n = a_1 r^{n-1} = 1 \cdot \left(\frac{1}{3}\right)^5 = \frac{1}{243}$.

3. What condition on consecutive differences identifies an arithmetic sequence?

   All $a_n - a_{n-1}$ are equal. Constant differences between consecutive terms characterize arithmetic sequences.

4. Identify the error: A student writes arithmetic explicit form as $a_n = a_1 + nd$. What is correct?

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

5. Given $a_3 = 11$ and $d = 4$ for an arithmetic sequence, find $a_1$.

   $a_1 = 3$. Use $a_3 = a_1 + 2d$ to solve: $11 = a_1 + 8$.

6. Find the next term in the geometric sequence $160, 80, 40, 20, \dots$.

   $10$. Multiply the last term $20$ by the common ratio $r = \frac{1}{2}$.

7. Write a recursive formula for the arithmetic sequence defined by $a_n = -2n + 9$.

   $a_1 = 7$; $a_n = a_{n-1} - 2$. Convert $a_n = -2n + 9$ to recursive by finding $a_1 = 7$ and $d = -2$.

8. Find $a_6$ for the arithmetic sequence with $a_1 = 4$ and $d = 3$.

   $a_6 = 19$. Use $a_n = a_1 + (n-1)d = 4 + 5(3) = 19$.

9. Find the missing term to make $4, x, 16$ an arithmetic sequence.

   $x = 10$. Use arithmetic mean: $x = \frac{4 + 16}{2} = 10$.

10. Model: A phone value is $900$ and loses $120$ each year. Write a recursive rule for $V_n$.

    $V_1 = 900$; $V_n = V_{n-1} - 120$. Recursive form models constant yearly decreases.

11. Find $r$ for a geometric sequence with $a_2 = 6$ and $a_5 = 162$.

    $r = 3$. Use $\frac{a_5}{a_2} = r^3$ to solve: $\frac{162}{6} = 27 = r^3$.

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

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

13. Given $a_1 = 6$ and $a_2 = 15$ for an arithmetic sequence, what is $d$?

    $d = 9$. Common difference: $d = a_2 - a_1 = 15 - 6 = 9$.

14. Find the common difference $d$ for the arithmetic sequence $7, 3, -1, -5, \dots$.

    $d = -4$. Subtract consecutive terms: $3 - 7 = -4$.

15. What is the common ratio $r$ in a geometric sequence in terms of consecutive terms $a_n$ and $a_{n-1}$?

    $r = \frac{a_n}{a_{n-1}}$. The common ratio is found by dividing consecutive terms.

16. Identify the sequence type if the terms change by multiplying by a constant each time.

    Geometric sequence. Multiplying by a constant creates a geometric sequence with that common ratio.

17. Find $a_1$ for the arithmetic sequence with $d = -7$ and $a_6 = 9$.

    $a_1 = 44$. Use $a_6 = a_1 + 5d$ to solve: $9 = a_1 - 35$.

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

    $a_n = a_1 + (n-1)d$. This formula adds $(n-1)$ multiples of the common difference $d$ to the first term.

19. Find the missing term to make $4, x, 16$ a geometric sequence with positive ratio.

    $x = 8$. Use geometric mean: $x = \sqrt{4 \cdot 16} = 8$.

20. Given $a_n = 3\left(\frac{1}{2}\right)^{n-1}$, what are $a_1$ and $r$?

    $a_1 = 3$, $r = \frac{1}{2}$. Identify first term and common ratio from explicit form.

21. What is the common difference $d$ in an arithmetic sequence in terms of consecutive terms $a_n$ and $a_{n-1}$?

    $d = a_n - a_{n-1}$. The common difference is found by subtracting consecutive terms.

22. Translate $a_n = 7 + 6(n-1)$ into a recursive formula.

    $a_1 = 7$; $a_n = a_{n-1} + 6$. Convert explicit form to recursive by identifying $a_1$ and $d$.

23. Find the common ratio $r$ for the geometric sequence $2, 6, 18, 54, \dots$.

    $r = 3$. Divide consecutive terms: $\frac{6}{2} = 3$.

24. Find $a_8$ for the arithmetic sequence $a_1 = -5$ and $d = 2$.

    $a_8 = 9$. Use $a_n = a_1 + (n-1)d = -5 + 7(2) = 9$.

25. What is the geometric mean of two positive numbers $x$ and $y$ (the middle term in a geometric sequence)?

    $\sqrt{xy}$. The geometric mean is the square root of the product of two positive numbers.

26. Identify whether $-1, 2, 5, 8, \dots$ is arithmetic or geometric.

    Arithmetic. Differences are constant: $2-(-1) = 3$.

27. Translate $a_1 = 5$, $a_n = 3a_{n-1}$ into an explicit formula for $a_n$.

    $a_n = 5\cdot 3^{n-1}$. Use $a_n = a_1 r^{n-1}$ with $r = 3$.

28. Write $a_{n-1}$ in terms of $a_n$ and $r$ for a geometric sequence (assume $r \ne 0$).

    $a_{n-1} = \frac{a_n}{r}$. Rearrange the geometric sequence relationship.

29. Identify the error: A student writes geometric explicit form as $a_n = a_1 r^n$. What is correct?

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

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

    $S_n = 40000 + 1500(n-1)$. Arithmetic sequence models constant yearly increases.

31. Write an explicit formula for the arithmetic sequence $5, 9, 13, 17, \dots$.

    $a_n = 5 + 4(n-1)$. First term is $5$ and common difference is $4$.

32. Translate $a_n = 9\left(-2\right)^{n-1}$ into a recursive formula.

    $a_1 = 9$; $a_n = -2a_{n-1}$. Convert explicit form to recursive by identifying $a_1$ and $r$.

33. Translate $a_1 = 10$, $a_n = a_{n-1} - 2$ into an explicit formula for $a_n$.

    $a_n = 10 - 2(n-1)$. Use $a_n = a_1 + (n-1)d$ with $d = -2$.

34. Find $d$ for an arithmetic sequence with $a_2 = 9$ and $a_5 = 21$.

    $d = 4$. Use $a_5 - a_2 = 3d$ to solve: $21 - 9 = 12 = 3d$.

35. Write a recursive formula for the geometric sequence $81, 27, 9, 3, \dots$.

    $a_1 = 81$; $a_n = \frac{1}{3}a_{n-1}$. First term is $81$ and each term is $\frac{1}{3}$ times the previous.

36. Write $a_{n-1}$ in terms of $a_n$ and $d$ for an arithmetic sequence.

    $a_{n-1} = a_n - d$. Rearrange the arithmetic sequence relationship.

37. Given $a_1 = 6$ and $a_2 = 15$ for a geometric sequence, what is $r$?

    $r = \frac{5}{2}$. Common ratio: $r = \frac{a_2}{a_1} = \frac{15}{6} = \frac{5}{2}$.

38. Write an explicit formula for the geometric sequence $81, 27, 9, 3, \dots$.

    $a_n = 81\left(\frac{1}{3}\right)^{n-1}$. First term is $81$ and common ratio is $\frac{1}{3}$.

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

    $a_1$ given; $a_n = r a_{n-1}$. Each term equals the previous term multiplied by the common ratio $r$.

40. Identify the sequence type if the terms change by adding a constant each time.

    Arithmetic sequence. Adding a constant creates an arithmetic sequence with that common difference.

41. What is the arithmetic mean of two numbers $x$ and $y$ (the middle term in an arithmetic sequence)?

    $\frac{x+y}{2}$. The arithmetic mean is the average of two numbers.

42. Identify whether $3, -6, 12, -24, \dots$ is arithmetic or geometric.

    Geometric. Ratios are constant: $\frac{-6}{3} = -2$.

43. Given $a_4 = 54$ and $r = 3$ for a geometric sequence, find $a_1$.

    $a_1 = 2$. Use $a_4 = a_1 r^3$ to solve: $54 = a_1 \cdot 27$.

44. Find $a_1$ for the geometric sequence with $r = 2$ and $a_6 = 96$.

    $a_1 = 3$. Use $a_6 = a_1 r^5$ to solve: $96 = a_1 \cdot 32$.

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

    $S_n = 40000 + 1500(n-1)$. Arithmetic sequence models constant yearly increases.

46. Find the next term in the arithmetic sequence $-4, -1, 2, 5, \dots$.

    $8$. Add the common difference $d = 3$ to the last term $5$.

47. Write a recursive formula for the arithmetic sequence $5, 9, 13, 17, \dots$.

    $a_1 = 5$; $a_n = a_{n-1} + 4$. First term is $5$ and each term adds $4$ to the previous.

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

    $a_n = a_1 r^{n-1}$. This formula multiplies the first term by $r$ raised to the $(n-1)$th power.

49. Model: A phone value is $900 and loses $120 each year. Write a recursive rule for $V_n$.

    $V_1 = 900$; $V_n = V_{n-1} - 120$. Recursive form models constant yearly decreases.

50. Write an explicit formula for the geometric sequence with $a_1 = -3$ and $r = -2$.

    $a_n = -3\left(-2\right)^{n-1}$. Use explicit form $a_n = a_1 r^{n-1}$ with given values.

51. Given $a_n = 12 + 5(n-1)$, what are $a_1$ and $d$?

    $a_1 = 12$, $d = 5$. Identify first term and common difference from explicit form.

52. Model: A population starts at $500$ and grows by a factor of $1.08$ yearly. What is $P_n$?

    $P_n = 500\left(1.08\right)^{n-1}$. Geometric sequence models constant growth factor.

53. What condition on consecutive ratios identifies a geometric sequence?

    All $\frac{a_n}{a_{n-1}}$ are equal. Constant ratios between consecutive terms characterize geometric sequences.

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