AP Precalculus Flashcards: Change In Arithmetic And Geometric Sequences

What this deck covers

This deck focuses on Change In Arithmetic And Geometric Sequences, giving you a quick way to review the definitions, rules, and examples that matter most for AP Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Find the common difference: $7, 10, 13, 16, \text{...}$

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

Flashcard 2: State the formula for the sum of the first $n$ terms of an arithmetic sequence.

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

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

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

Flashcard 4: What is the first term if $a_{5} = 21$ and $d = 5$?

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

Flashcard 5: Find the common difference: $7, 10, 13, 16, \text{...}$

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

Flashcard 6: What type of sequence is $a_n = 3 \times 2^n$?

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

Flashcard 7: What type of sequence is $a_n = 5n$?

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

Flashcard 8: Find the number of terms in the geometric sequence: $1, 3, 9, \text{...}, 81$

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

Flashcard 9: How many terms are in the arithmetic sequence: $2, 5, 8, \text{...}, 20$?

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

Flashcard 10: What is the first term if $a_{5} = 21$ and $d = 5$?

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

Flashcard 11: What is $a_{10}$ if $a_1 = 3$ and $d = 2$ in an arithmetic sequence?

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

Flashcard 12: What is the sum of the first 4 terms of the sequence: $2, 4, 8, 16, \text{...}$?

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

Flashcard 13: Find the number of terms in the geometric sequence: $1, 3, 9, \text{...}, 81$

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

Flashcard 14: Identify the 4th term of the sequence: $3, 9, 27, \text{...}$

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

Flashcard 15: Describe the sequence $7, 21, 63, \text{...}$

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

Flashcard 16: Which type of sequence is defined by $a_n = 3n + 2$?

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

Flashcard 17: Calculate the 7th term: $4, 8, 12, 16, \text{...}$

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

Flashcard 18: State the formula for the $n$th term of a geometric sequence.

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

Flashcard 19: Find the common ratio: $5, 15, 45, \text{...}$

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

Flashcard 20: Identify the 5th term of the sequence: $2, 5, 8, 11, \text{...}$

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

Flashcard 21: If $a_1 = 2$ and $r = 4$, what is the $n$th term of the sequence?

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

Flashcard 22: What type of sequence is $a_n = 5n$?

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

Flashcard 23: State the formula for the sum of the first $n$ terms of a geometric sequence.

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

Flashcard 24: Which type of sequence is defined by $a_n = 3n + 2$?

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

Flashcard 25: State the formula for the $n$th term of an arithmetic sequence.

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

Flashcard 26: Determine the 6th term: $1, 4, 7, 10, \text{...}$

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

Flashcard 27: What is the first term if $a_{4} = 81$ and $r = 3$?

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

Flashcard 28: What is $a_{10}$ if $a_1 = 3$ and $d = 2$ in an arithmetic sequence?

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

Flashcard 29: Which type of sequence is defined by $a_n = 4 \times 2^{n-1}$?

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

Flashcard 30: Identify the 5th term of the sequence: $2, 5, 8, 11, \text{...}$

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

Flashcard 31: Find the sum of the first 4 terms: $2, 6, 18, \text{...}$

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

Flashcard 32: Find the sum of the first 4 terms: $2, 6, 18, \text{...}$

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

Flashcard 33: State the formula for the $n$th term of a geometric sequence.

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

Flashcard 34: Find the common ratio: $5, 15, 45, \text{...}$

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

Flashcard 35: Find the sum of the first 6 terms: $3, 6, 9, 12, \text{...}$

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

Flashcard 36: What is the common difference in an arithmetic sequence?

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

Flashcard 37: Find the $n$th term if $a_1 = 6$ and $d = 4$.

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

Flashcard 38: What is the sum of the first 5 terms of the sequence: $1, 3, 5, 7, \text{...}$?

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

Flashcard 39: Calculate the sum: $2, 6, 18, \text{...}, 162$

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

Flashcard 40: Describe the sequence $9, 14, 19, 24, \text{...}$

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

Flashcard 41: Identify the sequence: $5, 9, 13, 17, \text{...}$

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

Flashcard 42: Identify the sequence: $5, 9, 13, 17, \text{...}$

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

Flashcard 43: What is the common ratio in a geometric sequence?

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

Flashcard 44: What is the first term if $a_{4} = 81$ and $r = 3$?

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

Flashcard 45: If $a_1 = 5$ and $d = 3$, what is the $n$th term of the sequence?

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

Flashcard 46: Calculate the 5th term: $10, 30, 90, \text{...}$

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

Flashcard 47: If $a_1 = 5$ and $d = 3$, what is the $n$th term of the sequence?

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

Flashcard 48: Find the sum of the first 6 terms: $3, 6, 9, 12, \text{...}$

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

Flashcard 49: State the formula for the sum of the first $n$ terms of an arithmetic sequence.

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

Flashcard 50: If $a_1 = 2$ and $r = 4$, what is the $n$th term of the sequence?

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

Flashcard 51: What is the common ratio in a geometric sequence?

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

Flashcard 52: What is the common difference in an arithmetic sequence?

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

Flashcard 53: What is the sum of the first 4 terms of the sequence: $2, 4, 8, 16, \text{...}$?

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

Flashcard 54: Determine the 3rd term: $8, 24, 72, \text{...}$

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

Flashcard 55: What type of sequence is $a_n = 3 \times 2^n$?

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

Flashcard 56: What is the sum of the first 5 terms of the sequence: $1, 3, 5, 7, \text{...}$?

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

Flashcard 57: Calculate the 5th term: $10, 30, 90, \text{...}$

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

Flashcard 58: Calculate the 7th term: $4, 8, 12, 16, \text{...}$

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

Flashcard 59: Determine the 3rd term: $8, 24, 72, \text{...}$

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

Flashcard 60: Which type of sequence is defined by $a_n = 4 \times 2^{n-1}$?

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

Flashcard 61: State the formula for the sum of the first $n$ terms of a geometric sequence.

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

Flashcard 62: Find the $n$th term if $a_1 = 7$ and $r = 2$.

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

Flashcard 63: State the formula for the $n$th term of an arithmetic sequence.

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

Flashcard 64: Determine the 6th term: $1, 4, 7, 10, \text{...}$

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

Flashcard 65: Describe the sequence $9, 14, 19, 24, \text{...}$

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

Flashcard 66: Calculate the sum: $2, 6, 18, \text{...}, 162$

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

Flashcard 67: Identify the sequence: $10, 20, 40, 80, \text{...}$

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

Flashcard 68: Find the $n$th term if $a_1 = 6$ and $d = 4$.

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

Flashcard 69: Identify the sequence: $10, 20, 40, 80, \text{...}$

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

Flashcard 70: Find the $n$th term if $a_1 = 7$ and $r = 2$.

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

Flashcard 71: Describe the sequence $7, 21, 63, \text{...}$

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

Flashcard 72: How many terms are in the arithmetic sequence: $2, 5, 8, \text{...}, 20$?

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