Multiple Choice

1. In the notation used for sequences, what does $a_n$ represent?

   The initial value of the sequence.

   The common ratio of the sequence.

   The $n$-th term of the sequence.

   The rule defining the next term.

2. Which essential piece of information is required to begin generating terms using a recurrence relation?

   The starting term (initial condition, $a_1$ or $a_0$).

   The formula for the $n$-th term.

   The total number of terms ($n$).

   The sum of the first two terms.

3. What is the fundamental concept of a recurrence relation?

   Defining the next term ($a_{n+1}$) based solely on the current term ($a_n$).

   Defining the general formula for finding the sequence sum.

   Defining any term based on its position number ($n$).

   Defining a term by multiplying the initial term by the position number.

4. In the general recurrence relation format $a_{n+1} = \text{Rule involving } a_n$, what determines whether the sequence is arithmetic?

   The rule involves $a_n$ being squared.

   The rule involves a term being multiplied by $a_n$.

   The rule involves a constant value being added to or subtracted from $a_n$.

   The rule involves $a_n$ being divided by a constant.

5. A sequence where the relationship between consecutive terms is defined by a constant multiplier is called a/an:

   Arithmetic sequence.

   Geometric sequence.

   Linear sequence.

   Recursive sequence.

6. Consider the general recurrence relation: $a_{n+1} = 1.5 a_n$, with $a_1 = 10$. This sequence is classified as:

   Mixed, combining addition and multiplication.

   Arithmetic, with common difference $d = 1.5$.

   Geometric, with common ratio $r = 1.5$.

   Arithmetic, with common difference $d = 10$.

7. An arithmetic sequence starts at 12 and increases by 5 each step. Which recurrence relation correctly models this sequence?

   $a_{n+1} = 5 a_n$, $a_1 = 12$

   $a_{n+1} = a_n - 5$, $a_1 = 12$

   $a_{n+1} = a_n + 5$, $a_1 = 12$

   $a_{n+1} = a_n + 12$, $a_1 = 5$

8. Given the arithmetic recurrence relation $a_{n+1} = a_n + 9$ and $a_1 = 3$. What is the value of the third term, $a_3$?

   21

   30

   39

   12

9. An arithmetic sequence is defined by $a_1 = 40$ and $a_{n+1} = a_n - 6$. Calculate the fourth term, $a_4$.

   34

   22

   28

10. What is the common difference ($d$) implied by the recurrence relation $a_{n+1} = a_n + 15.5$?

    $d = 1.55$

    $d = 15.5$

    $d = a_n$

    $d = 15$

11. A company's revenue starts at \$25,000 and decreases by \$1,500 every quarter. If $a_n$ is the revenue in quarter $n$, what are the starting term and the common difference?

    $a_1 = 1,500$, $d = -25,000$

    $a_1 = 1,500$, $d = 25,000$

    $a_1 = 25,000$, $d = -1,500$

    $a_1 = 25,000$, $d = 1,500$

12. For the sequence $a_{n+1} = a_n - 10$ where $a_1 = 100$, what is the second term, $a_2$?

    $-10$

    90

    10

    110

13. If an arithmetic sequence has $a_1 = 0$ and $d = 15$, what is the value of $a_4$?

    60

    30

    45

    15

14. A geometric sequence starts at 6 and triples in value with each subsequent term. Which recurrence relation models this sequence?

    $a_{n+1} = a_n + 3$, $a_1 = 6$

    $a_{n+1} = 3 a_n$, $a_1 = 6$

    $a_{n+1} = 3 a_n + 6$

    $a_{n+1} = 6 a_n$, $a_1 = 3$

15. Given the geometric recurrence relation $a_{n+1} = 2 a_n$ and $a_1 = 5$. What is the value of the third term, $a_3$?

    10

    25

    15

    20

16. A geometric sequence is defined by $a_1 = 100$ and $a_{n+1} = 0.5 a_n$. Calculate the fourth term, $a_4$.

    12.5

    50

    6.25

    25

17. What is the common ratio ($r$) implied by the recurrence relation $a_{n+1} = 1.08 a_n$?

    $r = 10.8$

    $r = 0.08$

    $r = 1.08$

    $r = 108$

18. If an investment increases by 15% each year, what is the common ratio ($r$) used in the geometric recurrence relation?

    $r = 0.15$

    $r = 1.15$

    $r = 0.85$

    $r = 15$

19. A population starts at 5,000 individuals and declines by 5% per month. If $a_n$ is the population in month $n$, what is the value of $a_2$?

    4,500

    5,250

    4,750

20. A sequence begins 3, 12, 48, 192, ... What is the recurrence relation that defines this sequence?

    $a_{n+1} = 3 a_n$, $a_1 = 4$

    $a_{n+1} = 4 a_n$, $a_1 = 4$

    $a_{n+1} = 4 a_n$, $a_1 = 3$

    $a_{n+1} = a_n + 9$, $a_1 = 3$

21. A bank deposit of \$2,000 earns 4% compound interest annually. Which recurrence relation models the balance, $a_n$, after $n$ years?

    $a_{n+1} = 0.04 a_n$, $a_1 = 2000$

    $a_{n+1} = 4 a_n$, $a_1 = 2000$

    $a_{n+1} = a_n + 80$, $a_1 = 2000$

    $a_{n+1} = 1.04 a_n$, $a_1 = 2000$

22. When using the CAS calculator's standard (Run/Main) mode to generate terms recursively, what must you enter first to begin the process for $a_1 = 25$?

    The recursive rule (e.g., ANS $+ 3$)

    ANS

    25, followed by pressing EXECUTE.

    $a_{n+1}$

23. You are using the CAS calculator to model $a_{n+1} = a_n + 5$, starting with $a_1 = 10$. After entering 10 and pressing EXECUTE, you enter 'ANS + 5' and press EXECUTE once. What term have you just generated?

    $a_3$

    $a_1$

    $a_2$

    $a_0$

24. To find the value of $a_5$ for the sequence $a_1 = 50, a_{n+1} = a_n - 4$ using the CAS calculator's ANS feature, how many times must you press the EXECUTE button *after* inputting $a_1$?

    4 times

    6 times

    5 times

    3 times

25. A geometric sequence is defined by $a_1 = 10$ and $a_{n+1} = 2.5 a_n$. Using the CAS calculator technique, what is the value of $a_4$?

    25

    156.25

    62.5

    100

26. Using the CAS calculator, find the value of $a_5$ for the arithmetic sequence $a_1 = 5$ and $a_{n+1} = a_n + 10$.

    55

    35

    45

27. Which CAS calculator operation sequence correctly generates the terms for $a_1 = 400$ and $a_{n+1} = 0.75 a_n$?

    Enter 400, EXECUTE. Then ANS $* 0.75$, EXECUTE repeatedly.

    Enter 400, EXECUTE. Then ANS $* 400$, EXECUTE repeatedly.

    Enter 400, EXECUTE. Then ANS $+ 0.75$, EXECUTE repeatedly.

    Enter 0.75, EXECUTE. Then ANS $* 400$, EXECUTE repeatedly.

28. If you enter $a_1$ into your CAS calculator and then press EXECUTE with the recursive rule 8 times, which term are you generating with the final calculation?

    $a_9$

    $a_{10}$

    $a_8$

    $a_7$

29. Identify the type of sequence defined by $a_{n+1} = a_n - 5$, $a_1 = 80$.

    Arithmetic, with a common difference $d = -5$.

    Non-linear, because subtraction is involved.

    Geometric, with a ratio of 5.

    Geometric, because the terms decrease.

30. Identify the type of sequence defined by $a_{n+1} = 0.99 a_n$, $a_1 = 500$.

    Arithmetic, with $d = 0.99$.

    Geometric, with a common ratio $r = 0.99$.

    Linear, representing a small decline.

    Arithmetic, with a negative common difference.

31. Determine the recurrence relation for the sequence: 4, 12, 36, 108, ...

    $a_{n+1} = a_n + 8$, $a_1 = 4$

    $a_{n+1} = 2 a_n + 4$, $a_1 = 4$

    $a_{n+1} = 4 a_n$, $a_1 = 4$

    $a_{n+1} = 3 a_n$, $a_1 = 4$

32. Determine the recurrence relation for the sequence: 10, 6, 2, -2, ...

    $a_{n+1} = 0.6 a_n$, $a_1 = 10$

    $a_{n+1} = a_n + 4$, $a_1 = 10$

    $a_{n+1} = a_n - 10$, $a_1 = 4$

    $a_{n+1} = a_n - 4$, $a_1 = 10$

33. A loan balance starts at \$5,000. If \$500 is repaid each month, the balance follows the relation $a_{n+1} = a_n - 500$, $a_1 = 5000$. What is the loan balance after two repayments (i.e., the value of $a_3$)?

    \$4,000

    \$4,500

    \$500

34. A piece of equipment valued initially at \$10,000 depreciates by 10% each year. The recurrence relation is $a_{n+1} = 0.90 a_n$, $a_1 = 10000$. What is the value of the equipment after one year (i.e., $a_2$)?

    \$9,900

    \$10,900

    \$1,000

    \$9,000

35. A sequence is defined by the non-standard relation $a_1 = 3$ and $a_{n+1} = 2 a_n + 1$. Calculate the value of $a_3$.

    10

    7

    15

    13

36. An arithmetic sequence is defined by $a_1 = 1$ and $a_{n+1} = a_n + 0.5$. Using the CAS technique, what is the 7th term ($a_7$)?

    3.5

    4.0

    3.0

    4.5

37. A rare painting initially costs \$50,000 and increases its value by 4% each year. Which recurrence relation correctly models this growth, where $a_n$ is the value after $n-1$ years?

    $a_{n+1} = a_n + 4000$, $a_1 = 50000$

    $a_{n+1} = a_n + 0.04$, $a_1 = 50000$

    $a_{n+1} = 1.04 a_n$, $a_1 = 50000$

    $a_{n+1} = 1.4 a_n$, $a_1 = 50000$

38. For the sequence $a_1 = 50$ and $a_{n+1} = a_n - 7$, you enter 50, press EXECUTE, then enter 'ANS - 7'. What term's value is generated when you press EXECUTE for the 5th time?

    $a_5$ (Value: 22)

    $a_7$ (Value: 8)

    $a_4$ (Value: 29)

    $a_6$ (Value: 15)