Write Explicit or Recursive Functions - Algebra 1

1. What does the exponent $n-1$ represent in $a_n=a_1\cdot r^{n-1}$?

   The number of times the ratio $r$ is applied from term $1$ to term $n$. Shows how many times we multiply by $r$ to get from $a_1$ to $a_n$.

2. What is the recursive formula for "start with $9$ and multiply by $-2$ each step," using $a_1=9$?

   $a_1=9$, $a_n=-2\cdot a_{n-1}$. Base case and rule for multiplying by -2 each step.

3. What is the recursive formula for "start with $1000$ and increase by $8\%$ each step," using $P(0)$?

   $P(0)=1000$, $P(n)=1.08\cdot P(n-1)$. Base case and rule for 8% growth each step.

4. 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 geometric sequence formula with first term and common ratio.

5. What is the explicit formula if you start with $P_0$ and decrease by $p\%$ each step for $n$ steps?

   $P(n)=P_0\left(1-\frac{p}{100}\right)^n$. Convert percentage decrease to decimal and subtract from 1.

6. What is the recursive formula for a geometric sequence with common ratio $r$?

   $a_1$ given, and $a_n=r\cdot a_{n-1}$ for $n\ge 2$. Each term multiplies the previous term by the common ratio.

7. 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 arithmetic sequence formula with first term and common difference.

8. What is the common ratio $r$ for the geometric sequence $2,6,18,54,\dots$?

   $r=3$. Each term is tripled: $6÷2=3$, $18÷6=3$.

9. What is the explicit formula for an arithmetic sequence with $a_1=7$ and $d=4$?

   $a_n=7+4(n-1)$. Arithmetic sequence with first term 7 and common difference 4.

10. What is the explicit expression for "start with $200$ and triple each hour" after $n$ hours?

    $P(n)=200\cdot 3^n$. Exponential function with initial value 200 and growth factor 3.

11. What is the explicit formula for the arithmetic sequence $3,8,13,18,\dots$?

    $a_n=3+5(n-1)$. Arithmetic sequence formula with $a_1=3$ and $d=5$.

12. What is the explicit expression for "start with $12$ and increase by $5$ each step" after $n$ steps?

    $P(n)=12+5n$. Linear function starting at 12 with slope 5.

13. Which model fits "population increases by $300$ each year": linear or exponential?

    Linear. Adding a constant amount each year creates linear growth.

14. What is the recursive rule for "start with $500$ and keep $90\%$ each month" using $P(0)$?

    $P(0)=500$, $P(n)=0.9\cdot P(n-1)$. Base case and rule for keeping 90% each step.

15. What is the recursive formula for a value starting at $90$ and decreasing by $12\%$ each year?

    $V(0)=90$, $V(n)=0.88\cdot V(n-1)$. Base case and rule for multiplying by 0.88 each year.

16. What is the definition of an explicit formula for a sequence $a_n$?

    A rule that gives $a_n$ directly from $n$ (no prior terms needed). You can find any term directly using just the position $n$.

17. What is the explicit expression for "start with $80$ and decrease by $3$ each day" after $n$ days?

    $P(n)=80-3n$. Linear function starting at 80 with negative slope -3.

18. What is the explicit formula for "start with $9$ and multiply by $-2$ each step," using $a_1=9$?

    $a_n=9\cdot (-2)^{n-1}$. Geometric sequence with $a_1=9$ and common ratio -2.

19. What is the recursive formula for a geometric sequence with $a_1=6$ and $r=2$?

    $a_1=6$, $a_n=2\cdot a_{n-1}$. Base case and rule for doubling each previous term.

20. What is the explicit formula for the geometric sequence $2,6,18,54,\dots$?

    $a_n=2\cdot 3^{n-1}$. Geometric sequence formula with $a_1=2$ and $r=3$.

21. What is the recursive rule for "start with $200$ and triple each hour" using $P(0)$?

    $P(0)=200$, $P(n)=3\cdot P(n-1)$. Base case and rule for tripling each step.

22. Which phrase in context usually signals an arithmetic pattern: "adds a fixed amount" or "multiplies by a fixed factor"?

    "Adds a fixed amount". Constant addition indicates arithmetic/linear growth patterns.

23. What is the growth factor $b$ for an increase of $8\%$ per step?

    $b=1.08$. 8% increase means multiply by $1+0.08=1.08$.

24. What is the explicit formula if a taxi charges $4$ dollars plus $2$ dollars per mile for $m$ miles?

    $C(m)=4+2m$. Fixed cost of $4$ plus variable cost of $2$ per mile.

25. What is the common difference $d$ for the arithmetic sequence $3,8,13,18,\dots$?

    $d=5$. Each term increases by 5: $8-3=5$, $13-8=5$.

26. What is the explicit formula for "start with $50$ and add $7$ each week," written as $a_n$ with $a_1=50$?

    $a_n=50+7(n-1)$. Arithmetic sequence with $a_1=50$ and common difference 7.

27. What is the explicit formula for a value starting at $90$ and decreasing by $12\%$ each year for $n$ years?

    $V(n)=90\cdot (0.88)^n$. Exponential decay with 12% decrease means factor of 0.88.

28. What is the explicit formula for a geometric sequence with $a_1=6$ and $r=2$?

    $a_n=6\cdot 2^{n-1}$. Geometric sequence with first term 6 and common ratio 2.

29. What is the explicit formula for linear growth starting at $P_0$ with change $m$ per step ($n$ steps)?

    $P(n)=P_0+mn$. Linear function with initial value and constant rate of change.

30. Which model fits "population increases by $4\%$ each year": linear or exponential?

    Exponential. Multiplying by a constant percent each year creates exponential growth.

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

    $d=a_n-a_{n-1}$. The common difference is the constant amount added each step.

32. What is the recursive formula for an arithmetic sequence with $a_1=7$ and $d=4$?

    $a_1=7$, $a_n=a_{n-1}+4$. Base case and rule for adding 4 to each previous term.

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

    $r=\frac{a_n}{a_{n-1}}$. The common ratio is the constant factor multiplied each step.

34. What is the explicit formula for a geometric sequence with $a_1=10$ and $r=\frac{1}{2}$?

    $a_n=10\left(\frac{1}{2}\right)^{n-1}$. Geometric sequence with first term 10 and common ratio 1/2.

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

    $a_n=-3-2(n-1)$. Arithmetic sequence with first term -3 and common difference -2.

36. What is the definition of a recursive formula for a sequence $a_n$?

    A rule that gives $a_n$ using earlier term(s), plus a starting value. Each term depends on the previous term(s), so you build sequentially.

37. What is the explicit formula for earnings of $15$ dollars per hour for $h$ hours plus a $20$ dollar bonus?

    $E(h)=15h+20$. Variable earnings of $15 per hour plus fixed bonus of $20.

38. What is the recursive formula for an arithmetic sequence with common difference $d$?

    $a_1$ given, and $a_n=a_{n-1}+d$ for $n\ge 2$. Each term adds the common difference to the previous term.

39. What is the explicit formula if you start with $P_0$ and increase by $p\%$ each step for $n$ steps?

    $P(n)=P_0\left(1+\frac{p}{100}\right)^n$. Convert percentage increase to decimal and add to 1.

40. What two parts must a recursive definition of $a_n$ always include?

    A starting term (like $a_1$) and a recursion rule (like $a_n=a_{n-1}+d$). You need a base case and a pattern to find the next term.

41. What does $n-1$ represent in the arithmetic explicit formula $a_n=a_1+(n-1)d$?

    The number of steps from term $1$ to term $n$. Shows how many times we add $d$ to get from $a_1$ to $a_n$.

42. What is the decay factor $b$ for a decrease of $35\%$ per step?

    $b=0.65$. 35% decrease means multiply by $1-0.35=0.65$.

43. What is the recursive formula for "start with $50$ and add $7$ each week," written as $a_n$ with $a_1=50$?

    $a_1=50$, $a_n=a_{n-1}+7$. Base case and rule for adding 7 to each previous term.

44. What is the explicit formula for a balance starting at $120$ and increasing by $6\%$ each month for $n$ months?

    $B(n)=120\cdot (1.06)^n$. Exponential growth with 6% increase means factor of 1.06.

45. What is the explicit expression for "start with $500$ and keep $90\%$ each month" after $n$ months?

    $P(n)=500\cdot (0.9)^n$. Exponential decay with initial value 500 and decay factor 0.9.

46. What is the recursive formula for linear growth starting at $P_0$ with change $m$ per step?

    $P(0)=P_0$, and $P(n)=P(n-1)+m$ for $n\ge 1$. Each step adds the constant change $m$ to the previous value.

47. What is the explicit formula for exponential growth starting at $P_0$ with factor $b$ per step?

    $P(n)=P_0\cdot b^n$. Exponential function with initial value and constant growth factor.

48. What is the recursive rule for "start with $12$ and increase by $5$ each step" using $P(0)$?

    $P(0)=12$, $P(n)=P(n-1)+5$. Base case and rule for adding 5 each step.

49. What is the recursive formula for exponential growth starting at $P_0$ with factor $b$ per step?

    $P(0)=P_0$, and $P(n)=b\cdot P(n-1)$ for $n\ge 1$. Each step multiplies the previous value by the growth factor $b$.

50. What is the recursive rule for "start with $80$ and decrease by $3$ each day" using $P(0)$?

    $P(0)=80$, $P(n)=P(n-1)-3$. Base case and rule for subtracting 3 each step.

51. What is the recursive formula for "start with $1000$ and decrease by $35\%$ each step," using $P(0)$?

    $P(0)=1000$, $P(n)=0.65\cdot P(n-1)$. Base case and rule for 35% decay each step.

52. Which phrase in context usually signals a geometric pattern: "adds a fixed amount" or "multiplies by a fixed factor"?

    "Multiplies by a fixed factor". Constant multiplication indicates geometric/exponential growth patterns.

53. What is the recursive formula for the sequence $2,6,18,54,\dots$ using $a_1$?

    $a_1=2$, $a_n=3\cdot a_{n-1}$. Base case and rule for tripling each previous term.

54. What is the recursive formula for a balance starting at $120$ and increasing by $6\%$ each month?

    $B(0)=120$, $B(n)=1.06\cdot B(n-1)$. Base case and rule for multiplying by 1.06 each month.

Study these flashcards