Change in Arithmetic and Geometric Sequences - AP Precalculus

A savings account grows geometrically with $a_1=500$ and $r=1.10$ : 500; 550; 605. Based on the sequence, what is $a_6$ ?
Answer: $805.26$ 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.
A salary follows an arithmetic sequence with $a_1=50{,}000$ and $d=2{,}000$ : 50,000; 52,000; 54,000. Based on the sequence, what is the $n$ th term $a_n$ ?
Answer: $a_n=50{,}000+2{,}000(n-1)$ 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.
Bacteria doubles hourly in a geometric sequence with $a_1=200$ and $r=2$ : 200; 400; 800. Using the given terms, what is the $n$ th term of the sequence?
Answer: $a_n=200\cdot 2^{n-1}$ 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.
A city's population declines geometrically with $a_1=100{,}000$ and $r=0.98$ : 100,000; 98,000; 96,040. Based on the sequence, how does the sequence change from term 1 to term 4?
Answer: Decreases by $5{,}880.80$ 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.
A city's population declines geometrically with $a_1=100{,}000$ and $r=0.95$ : 100,000; 95,000; 90,250. Using the given terms, what is $a_5$ ?
Answer: $81{,}450.63$ 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.
Bacteria doubles hourly in a geometric sequence with $a_1=50$ and $r=2$ : 50; 100; 200. Using the given terms, how does the sequence change from term 3 to term 6?
Answer: Increases by $1{,}400$ 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.
A city's population declines geometrically with $a_1=80{,}000$ and $r=0.90$ : 80,000; 72,000; 64,800. Using the given terms, determine the sum of the first 4 terms.
Answer: $275{,}120$ 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.
A salary follows an arithmetic sequence with $a_1=30{,}000$ and $d=1{,}000$ : 30,000; 31,000; 32,000. Based on the sequence, calculate the total change after 10 terms.
Answer: $9{,}000$ 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.
A salary follows an arithmetic sequence with $a_1=30{,}000$ and $d=1{,}000$ : 30,000; 31,000; 32,000. Based on the sequence, how does the sequence change from term 2 to term 8?
Answer: Increases by $6{,}000$ 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).
A salary follows an arithmetic sequence with $a_1=50{,}000$ and $d=2{,}000$ : 50,000; 52,000; 54,000. Based on the sequence, determine the sum of the first 5 terms.
Answer: $270{,}000$ 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.
A city's population declines geometrically with $a_1=50{,}000$ and $r=0.95$ : 50,000; 47,500; 45,125. Based on the sequence, what is the $n$ th term $a_n$ ?
Answer: $a_n=50{,}000(0.95)^{n-1}$ 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.
Bacteria doubles hourly in a geometric sequence with $a_1=200$ and $r=2$ : 200; 400; 800. Using the given terms, what is $a_7$ ?
Answer: $12{,}800$ 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.
A savings account grows geometrically with $a_1=1{,}000$ and $r=1.05$ : 1,000; 1,050; 1,102.50. Using the given terms, what is $a_5$ ?
Answer: $1{,}215.51$ 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.

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