AP Precalculus Quiz: Change In Arithmetic And Geometric Sequences

What this quiz covers

This quiz focuses on Change In Arithmetic And Geometric Sequences, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Precalculus.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. $a_n=200n$
2. $a_n=200\cdot 2^{n-1}$ (correct answer)
3. $a_n=200\cdot 2^{n}$
4. $a_n=200+(n-1)2$

Explanation: This question tests understanding of changes in arithmetic and geometric sequences, crucial for AP Precalculus. Arithmetic sequences change by a fixed difference, while geometric sequences change by a fixed ratio. In this scenario, the sequence starts with 200 bacteria, doubling each hour (ratio of 2). Choice B is correct because it accurately represents the geometric sequence formula: aₙ = a₁ × r^(n-1) = 200 × 2^(n-1). Choice C (200 × 2ⁿ) is incorrect due to using n instead of (n-1) as the exponent, a common mistake when students forget that the first term uses r⁰. To help students: Remember that geometric sequences use r^(n-1) because the first term has no multiplication by r. Practice writing out the first few terms to verify the formula pattern.

Question 2

1. $9{,}000$ (correct answer)
2. $10{,}000$
3. $8{,}000$
4. $11{,}000$

Explanation: This question tests understanding of changes in arithmetic and geometric sequences, crucial for AP Precalculus. Arithmetic sequences change by a fixed difference, while geometric sequences change by a fixed ratio. In this scenario, the sequence starts with 30,000, changing by a difference of 1,000 each term. Choice A is correct because the total change after 10 terms means finding the difference between a₁₀ and a₁: a₁₀ = 30,000 + 9(1,000) = 39,000, so the change is 39,000 - 30,000 = 9,000. Choice B (10,000) is incorrect due to confusing the number of terms with the number of differences, a common mistake when students forget that reaching the 10th term requires only 9 steps. To help students: Remember that the change from term 1 to term n involves (n-1) differences. Draw out the first few terms to visualize the pattern.

Question 3

1. Increases by $1{,}400$ (correct answer)
2. Increases by $200$
3. Decreases by $1{,}400$
4. Increases by $700$

Explanation: This question tests understanding of changes in arithmetic and geometric sequences, crucial for AP Precalculus. Arithmetic sequences change by a fixed difference, while geometric sequences change by a fixed ratio. In this scenario, the sequence starts with 50 bacteria, doubling each hour. Choice A is correct because it accurately calculates the change: a_3 = 200 and a_6 = 50 × 2^5 = 50 × 32 = 1,600, giving an increase of 1,600 - 200 = 1,400. Choice B is incorrect with only 200, likely confusing the value of a_3 with the change between terms. To help students: Practice finding specific terms before calculating differences. Emphasize the importance of reading carefully to identify which terms are being compared.

Question 4

1. $a_n=50{,}000+2{,}000n$
2. $a_n=50{,}000(2{,}000)^{n-1}$
3. $a_n=50{,}000+2{,}000(n-1)$ (correct answer)
4. $a_n=50{,}000+1{,}000(n-1)$

Explanation: This question tests understanding of changes in arithmetic and geometric sequences, crucial for AP Precalculus. Arithmetic sequences change by a fixed difference, while geometric sequences change by a fixed ratio. In this scenario, the sequence starts with 50,000, changing by a difference of 2,000 each term. Choice C is correct because it accurately applies the arithmetic sequence formula a_n = a_1 + d(n-1), giving a_n = 50,000 + 2,000(n-1). Choice A is incorrect due to using n instead of (n-1), a common mistake when students forget that the first term already includes a_1. To help students: Practice identifying the correct formula structure for arithmetic sequences. Emphasize that (n-1) represents the number of times we add the common difference, starting from the first term.

Question 5

1. $260{,}000$
2. $270{,}000$ (correct answer)
3. $280{,}000$
4. $156{,}000$

Explanation: This question tests understanding of changes in arithmetic and geometric sequences, crucial for AP Precalculus. Arithmetic sequences change by a fixed difference, while geometric sequences change by a fixed ratio. In this scenario, the sequence starts with 50,000, changing by a difference of 2,000 each term. Choice B is correct because it accurately applies the arithmetic series formula: S₅ = 5/2 × (2a₁ + 4d) = 5/2 × (100,000 + 8,000) = 5/2 × 108,000 = 270,000. Choice A (260,000) is incorrect due to miscalculating the sum formula, a common mistake when students forget to properly apply the arithmetic series formula. To help students: Practice using the sum formula S_n = n/2 × (2a₁ + (n-1)d) systematically. Emphasize verifying calculations by adding the first few terms manually as a check.

Question 6

1. $12{,}800$ (correct answer)
2. $6{,}400$
3. $1{,}400$
4. $25{,}600$

Explanation: This question tests understanding of changes in arithmetic and geometric sequences, crucial for AP Precalculus. Arithmetic sequences change by a fixed difference, while geometric sequences change by a fixed ratio. In this scenario, the sequence starts with 200 bacteria, doubling (ratio of 2) each hour. Choice A is correct because it accurately applies the geometric sequence formula a_n = a_1 × r^(n-1), giving a_7 = 200 × 2^6 = 200 × 64 = 12,800. Choice B is incorrect due to calculating 2^5 instead of 2^6, a common mistake when students forget that the exponent is (n-1). To help students: Practice counting the number of multiplications needed to reach the nth term. Emphasize that for the 7th term, we multiply by r exactly 6 times.

Question 7

1. $a_n=50{,}000+0.95(n-1)$
2. $a_n=50{,}000(0.95)^{n-1}$ (correct answer)
3. $a_n=50{,}000(1.05)^{n-1}$
4. $a_n=50{,}000-0.95(n-1)$

Explanation: This question tests understanding of changes in arithmetic and geometric sequences, crucial for AP Precalculus. Arithmetic sequences change by a fixed difference, while geometric sequences change by a fixed ratio. In this scenario, the sequence starts with 50,000, declining by a ratio of 0.95 each term. Choice B is correct because it accurately applies the geometric sequence formula a_n = a_1 × r^(n-1), giving a_n = 50,000(0.95)^(n-1). Choice A is incorrect due to treating this as an arithmetic sequence with a difference of 0.95, a common mistake when students see a decimal and assume subtraction. To help students: Practice identifying sequence types from the given terms. Emphasize that ratios less than 1 still indicate geometric sequences, not arithmetic ones.

Question 8

1. Increases by $5{,}000$
2. Increases by $6{,}000$ (correct answer)
3. Increases by $7{,}000$
4. Decreases by $6{,}000$

Explanation: This question tests understanding of changes in arithmetic and geometric sequences, crucial for AP Precalculus. Arithmetic sequences change by a fixed difference, while geometric sequences change by a fixed ratio. In this scenario, the sequence starts with 30,000, increasing by 1,000 each term. Choice B is correct because in an arithmetic sequence, the change from term 2 to term 8 is 6 times the common difference: 6 × 1,000 = 6,000. Choice C is incorrect with 7,000, likely from counting 7 steps instead of 6, a common mistake when students include both endpoints. To help students: Practice counting the number of steps between terms carefully. Emphasize that from term 2 to term 8, there are exactly 6 steps (8 - 2 = 6).

Question 9

1. $81{,}450.63$ (correct answer)
2. $85{,}737.50$
3. $77{,}378.09$
4. $814{,}506.25$

Explanation: This question tests understanding of changes in arithmetic and geometric sequences, crucial for AP Precalculus. Arithmetic sequences change by a fixed difference, while geometric sequences change by a fixed ratio. In this scenario, the sequence starts with 100,000, changing by a ratio of 0.95 each term (representing a 5% decline). Choice A is correct because it accurately applies the geometric sequence formula: a₅ = a₁ × r⁴ = 100,000 × (0.95)⁴ = 100,000 × 0.81450625 = 81,450.63. Choice C (77,378.09) is incorrect due to using r⁵ instead of r⁴, a common mistake when students confuse the term number with the exponent. To help students: Remember that for the nth term, use r^(n-1) as the exponent. Practice calculating several terms to build confidence with the pattern.

Question 10

1. $805.26$ (correct answer)
2. $732.05$
3. $885.78$
4. $80.53$

Explanation: This question tests understanding of changes in arithmetic and geometric sequences, crucial for AP Precalculus. Arithmetic sequences change by a fixed difference, while geometric sequences change by a fixed ratio. In this scenario, the sequence starts with 500, growing by a ratio of 1.10 each term (10% growth). Choice A is correct because it accurately applies the geometric sequence formula: a₆ = a₁ × r⁵ = 500 × (1.10)⁵ = 500 × 1.61051 = 805.26. Choice B (732.05) is incorrect due to using the wrong power of r, a common mistake when students miscalculate compound growth. To help students: Remember that for the nth term, use r^(n-1). Practice with a calculator to ensure accuracy with decimal powers.

Question 11

1. $275{,}120$ (correct answer)
2. $295{,}200$
3. $268{,}920$
4. $27{,}512$

Explanation: This question tests understanding of changes in arithmetic and geometric sequences, crucial for AP Precalculus. Arithmetic sequences change by a fixed difference, while geometric sequences change by a fixed ratio. In this scenario, the sequence starts with 80,000, declining by a ratio of 0.90 each term. Choice A is correct because it accurately calculates the sum: a₁ = 80,000, a₂ = 72,000, a₃ = 64,800, a₄ = 58,320, giving a sum of 275,120. Choice C (268,920) is incorrect due to calculation errors in finding individual terms, a common mistake when students rush through geometric calculations. To help students: Calculate each term carefully before summing. For geometric series, you can also use the formula S_n = a₁(1-rⁿ)/(1-r) as a check.

Question 12

1. $1{,}215.51$ (correct answer)
2. $1{,}250.00$
3. $12{,}155.06$
4. $1{,}205.00$

Explanation: This question tests understanding of changes in arithmetic and geometric sequences, crucial for AP Precalculus. Arithmetic sequences change by a fixed difference, while geometric sequences change by a fixed ratio. In this scenario, the sequence starts with 1,000, changing by a ratio of 1.05 each term. Choice A is correct because it accurately applies the geometric sequence formula a_n = a_1 × r^(n-1), giving a_5 = 1,000 × 1.05^4 = 1,000 × 1.21550625 = 1,215.51. Choice C is incorrect due to multiplying by 10, likely from misreading the decimal point, a common mistake when students rush calculations. To help students: Practice careful calculation with geometric sequences, especially with decimal ratios. Emphasize checking that answers are reasonable given the growth rate.

Question 13

1. Decreases by $5{,}880.80$ (correct answer)
2. Decreases by $6{,}000.00$
3. Increases by $5{,}880.80$
4. Decreases by $5{,}880.08$

Explanation: This question tests understanding of changes in arithmetic and geometric sequences, crucial for AP Precalculus. Arithmetic sequences change by a fixed difference, while geometric sequences change by a fixed ratio. In this scenario, the sequence starts with 100,000, declining by a ratio of 0.98 each term. Choice A is correct because it accurately calculates the change from term 1 to term 4: a_4 = 100,000 × 0.98^3 = 94,119.20, giving a decrease of 100,000 - 94,119.20 = 5,880.80. Choice B is incorrect due to assuming a linear decrease of 2,000 per term, a common mistake when students confuse geometric and arithmetic sequences. To help students: Practice distinguishing between multiplicative and additive changes. Emphasize that geometric sequences with r < 1 show decreasing changes between consecutive terms.