Change in Arithmetic and Geometric Sequences - AP Precalculus

1. State the formula for the sum of the first $n$ terms of an arithmetic sequence.

   $S_n = \frac{n}{2} (a_1 + a_n)$. Uses the average of first and last terms, multiplied by number of terms.

2. What is the first term if $a_{5} = 21$ and $d = 5$?

   $1$. From $a_5 = a_1 + 4d = 21$, solve: $a_1 = 21 - 4(5) = 1$.

3. Find the common difference: $7, 10, 13, 16, \text{...}$

   $3$. Subtract consecutive terms: $10 - 7 = 3$.

4. What type of sequence is $a_n = 5n$?

   Arithmetic sequence. Linear form $5n$ indicates constant difference (here $d = 5$).

5. Find the number of terms in the geometric sequence: $1, 3, 9, \text{...}, 81$

   $5$. Solve $81 = 1 \times 3^{n-1}$ to find $n = 5$.

6. Identify the 4th term of the sequence: $3, 9, 27, \text{...}$

   $81$. Common ratio is $3$, so $a_4 = 3 \times 3^{4-1} = 3 \times 27 = 81$.

7. Describe the sequence $7, 21, 63, \text{...}$

   Geometric sequence. Each term is multiplied by $3$ (constant ratio).

8. Which type of sequence is defined by $a_n = 3n + 2$?

   Arithmetic sequence. Linear form $3n + 2$ indicates constant difference between consecutive terms.

9. Calculate the 7th term: $4, 8, 12, 16, \text{...}$

   $28$. Common difference is $4$, so $a_7 = 4 + (7-1) \times 4 = 28$.

10. State the formula for the $n$th term of a geometric sequence.

    $a_n = a_1 \times r^{n-1}$. Start with first term $a_1$, then multiply by $r$ raised to $(n-1)$ power.

11. Find the common ratio: $5, 15, 45, \text{...}$

    $3$. Divide consecutive terms: $15 \div 5 = 3$.

12. If $a_1 = 2$ and $r = 4$, what is the $n$th term of the sequence?

    $a_n = 2 \times 4^{n-1}$. Substitute given values into the geometric sequence formula.

13. State the formula for the $n$th term of an arithmetic sequence.

    $a_n = a_1 + (n-1)d$. Start with first term $a_1$, then add $(n-1)$ times the common difference.

14. Determine the 6th term: $1, 4, 7, 10, \text{...}$

    $16$. Common difference is $3$, so $a_6 = 1 + (6-1) \times 3 = 16$.

15. What is $a_{10}$ if $a_1 = 3$ and $d = 2$ in an arithmetic sequence?

    $21$. Use formula: $a_{10} = 3 + (10-1) \times 2 = 21$.

16. Identify the 5th term of the sequence: $2, 5, 8, 11, \text{...}$

    $14$. Common difference is $3$, so $a_5 = 2 + (5-1) \times 3 = 14$.

17. Find the sum of the first 4 terms: $2, 6, 18, \text{...}$

    $80$. Sum the terms: $2 + 6 + 18 + 54 = 80$.

18. What is the common difference in an arithmetic sequence?

    The constant difference between consecutive terms. Each term is found by adding this same value to the previous term.

19. Find the $n$th term if $a_1 = 6$ and $d = 4$.

    $a_n = 6 + (n-1) \times 4$. Substitute given values into the arithmetic sequence formula.

20. Describe the sequence $9, 14, 19, 24, \text{...}$

    Arithmetic sequence. Each term increases by $5$ (constant difference).

21. Identify the sequence: $5, 9, 13, 17, \text{...}$

    Arithmetic sequence. Each term adds $4$ to the previous term (constant difference).

22. What is the common ratio in a geometric sequence?

    The constant factor between consecutive terms. Each term is found by multiplying the previous term by this same value.

23. What is the first term if $a_{4} = 81$ and $r = 3$?

    $3$. From $a_4 = a_1 \times r^3 = 81$, solve: $a_1 = 81 \div 3^3 = 3$.

24. If $a_1 = 5$ and $d = 3$, what is the $n$th term of the sequence?

    $a_n = 5 + (n-1) \times 3$. Substitute given values into the arithmetic sequence formula.

25. Find the sum of the first 6 terms: $3, 6, 9, 12, \text{...}$

    $63$. Sum formula: $S_6 = \frac{6}{2}(3 + 18) = 3 \times 21 = 63$.

26. What is the sum of the first 4 terms of the sequence: $2, 4, 8, 16, \text{...}$?

    $30$. Add the terms: $2 + 4 + 8 + 16 = 30$.

27. What type of sequence is $a_n = 3 \times 2^n$?

    Geometric sequence. Exponential form $3 \times 2^n$ indicates constant ratio (here $r = 2$).

28. What is the sum of the first 5 terms of the sequence: $1, 3, 5, 7, \text{...}$?

    $25$. Add the terms: $1 + 3 + 5 + 7 + 9 = 25$.

29. Calculate the 5th term: $10, 30, 90, \text{...}$

    $810$. Common ratio is $3$, so $a_5 = 10 \times 3^{5-1} = 10 \times 81 = 810$.

30. Determine the 3rd term: $8, 24, 72, \text{...}$

    $72$. Given sequence shows $a_3 = 72$ directly from the pattern.

31. Which type of sequence is defined by $a_n = 4 \times 2^{n-1}$?

    Geometric sequence. Exponential form $4 \times 2^{n-1}$ indicates constant ratio between consecutive terms.

32. State the formula for the sum of the first $n$ terms of a geometric sequence.

    $S_n = a_1 \frac{1-r^n}{1-r}$, $r \neq 1$. Uses first term times the finite geometric series formula when $r \neq 1$.

33. Find the $n$th term if $a_1 = 7$ and $r = 2$.

    $a_n = 7 \times 2^{n-1}$. Substitute given values into the geometric sequence formula.

34. Calculate the sum: $2, 6, 18, \text{...}, 162$

    $242$. Use geometric sum formula with $a_1 = 2$, $r = 3$, and $n = 5$.

35. Identify the sequence: $10, 20, 40, 80, \text{...}$

    Geometric sequence. Each term doubles the previous term (constant ratio of $2$).

36. How many terms are in the arithmetic sequence: $2, 5, 8, \text{...}, 20$?

    $7$. Solve $20 = 2 + (n-1) \times 3$ to find $n = 7$.

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