Change in Arithmetic and Geometric Sequences - AP Precalculus
Card 1 of 30
What is the sum of the first 4 terms of the sequence: $2, 4, 8, 16, \text{...}$?
What is the sum of the first 4 terms of the sequence: $2, 4, 8, 16, \text{...}$?
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$30$. Add the terms: $2 + 4 + 8 + 16 = 30$.
$30$. Add the terms: $2 + 4 + 8 + 16 = 30$.
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State the formula for the $n$th term of a geometric sequence.
State the formula for the $n$th term of a geometric sequence.
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$a_n = a_1 \times r^{n-1}$. Start with first term $a_1$, then multiply by $r$ raised to $(n-1)$ power.
$a_n = a_1 \times r^{n-1}$. Start with first term $a_1$, then multiply by $r$ raised to $(n-1)$ power.
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Identify the 5th term of the sequence: $2, 5, 8, 11, \text{...}$
Identify the 5th term of the sequence: $2, 5, 8, 11, \text{...}$
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$14$. Common difference is $3$, so $a_5 = 2 + (5-1) \times 3 = 14$.
$14$. Common difference is $3$, so $a_5 = 2 + (5-1) \times 3 = 14$.
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Which type of sequence is defined by $a_n = 4 \times 2^{n-1}$?
Which type of sequence is defined by $a_n = 4 \times 2^{n-1}$?
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Geometric sequence. Exponential form $4 \times 2^{n-1}$ indicates constant ratio between consecutive terms.
Geometric sequence. Exponential form $4 \times 2^{n-1}$ indicates constant ratio between consecutive terms.
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Determine the 3rd term: $8, 24, 72, \text{...}$
Determine the 3rd term: $8, 24, 72, \text{...}$
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$72$. Given sequence shows $a_3 = 72$ directly from the pattern.
$72$. Given sequence shows $a_3 = 72$ directly from the pattern.
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Which type of sequence is defined by $a_n = 3n + 2$?
Which type of sequence is defined by $a_n = 3n + 2$?
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Arithmetic sequence. Linear form $3n + 2$ indicates constant difference between consecutive terms.
Arithmetic sequence. Linear form $3n + 2$ indicates constant difference between consecutive terms.
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Which type of sequence is defined by $a_n = 4 \times 2^{n-1}$?
Which type of sequence is defined by $a_n = 4 \times 2^{n-1}$?
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Geometric sequence. Exponential form $4 \times 2^{n-1}$ indicates constant ratio between consecutive terms.
Geometric sequence. Exponential form $4 \times 2^{n-1}$ indicates constant ratio between consecutive terms.
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Find the common difference: $7, 10, 13, 16, \text{...}$
Find the common difference: $7, 10, 13, 16, \text{...}$
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$3$. Subtract consecutive terms: $10 - 7 = 3$.
$3$. Subtract consecutive terms: $10 - 7 = 3$.
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What is the sum of the first 5 terms of the sequence: $1, 3, 5, 7, \text{...}$?
What is the sum of the first 5 terms of the sequence: $1, 3, 5, 7, \text{...}$?
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$25$. Add the terms: $1 + 3 + 5 + 7 + 9 = 25$.
$25$. Add the terms: $1 + 3 + 5 + 7 + 9 = 25$.
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State the formula for the sum of the first $n$ terms of an arithmetic sequence.
State the formula for the sum of the first $n$ terms of an arithmetic sequence.
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$S_n = \frac{n}{2} (a_1 + a_n)$. Uses the average of first and last terms, multiplied by number of terms.
$S_n = \frac{n}{2} (a_1 + a_n)$. Uses the average of first and last terms, multiplied by number of terms.
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State the formula for the sum of the first $n$ terms of a geometric sequence.
State the formula for the sum of the first $n$ terms of a geometric sequence.
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$S_n = a_1 \frac{1-r^n}{1-r}$, $r \neq 1$. Uses first term times the finite geometric series formula when $r \neq 1$.
$S_n = a_1 \frac{1-r^n}{1-r}$, $r \neq 1$. Uses first term times the finite geometric series formula when $r \neq 1$.
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Determine the 6th term: $1, 4, 7, 10, \text{...}$
Determine the 6th term: $1, 4, 7, 10, \text{...}$
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$16$. Common difference is $3$, so $a_6 = 1 + (6-1) \times 3 = 16$.
$16$. Common difference is $3$, so $a_6 = 1 + (6-1) \times 3 = 16$.
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Determine the 3rd term: $8, 24, 72, \text{...}$
Determine the 3rd term: $8, 24, 72, \text{...}$
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$72$. Given sequence shows $a_3 = 72$ directly from the pattern.
$72$. Given sequence shows $a_3 = 72$ directly from the pattern.
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What is $a_{10}$ if $a_1 = 3$ and $d = 2$ in an arithmetic sequence?
What is $a_{10}$ if $a_1 = 3$ and $d = 2$ in an arithmetic sequence?
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$21$. Use formula: $a_{10} = 3 + (10-1) \times 2 = 21$.
$21$. Use formula: $a_{10} = 3 + (10-1) \times 2 = 21$.
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What type of sequence is $a_n = 3 \times 2^n$?
What type of sequence is $a_n = 3 \times 2^n$?
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Geometric sequence. Exponential form $3 \times 2^n$ indicates constant ratio (here $r = 2$).
Geometric sequence. Exponential form $3 \times 2^n$ indicates constant ratio (here $r = 2$).
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What type of sequence is $a_n = 5n$?
What type of sequence is $a_n = 5n$?
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Arithmetic sequence. Linear form $5n$ indicates constant difference (here $d = 5$).
Arithmetic sequence. Linear form $5n$ indicates constant difference (here $d = 5$).
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Calculate the sum: $2, 6, 18, \text{...}, 162$
Calculate the sum: $2, 6, 18, \text{...}, 162$
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$242$. Use geometric sum formula with $a_1 = 2$, $r = 3$, and $n = 5$.
$242$. Use geometric sum formula with $a_1 = 2$, $r = 3$, and $n = 5$.
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Find the $n$th term if $a_1 = 7$ and $r = 2$.
Find the $n$th term if $a_1 = 7$ and $r = 2$.
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$a_n = 7 \times 2^{n-1}$. Substitute given values into the geometric sequence formula.
$a_n = 7 \times 2^{n-1}$. Substitute given values into the geometric sequence formula.
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Find the $n$th term if $a_1 = 6$ and $d = 4$.
Find the $n$th term if $a_1 = 6$ and $d = 4$.
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$a_n = 6 + (n-1) \times 4$. Substitute given values into the arithmetic sequence formula.
$a_n = 6 + (n-1) \times 4$. Substitute given values into the arithmetic sequence formula.
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Find the number of terms in the geometric sequence: $1, 3, 9, \text{...}, 81$
Find the number of terms in the geometric sequence: $1, 3, 9, \text{...}, 81$
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$5$. Solve $81 = 1 \times 3^{n-1}$ to find $n = 5$.
$5$. Solve $81 = 1 \times 3^{n-1}$ to find $n = 5$.
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How many terms are in the arithmetic sequence: $2, 5, 8, \text{...}, 20$?
How many terms are in the arithmetic sequence: $2, 5, 8, \text{...}, 20$?
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$7$. Solve $20 = 2 + (n-1) \times 3$ to find $n = 7$.
$7$. Solve $20 = 2 + (n-1) \times 3$ to find $n = 7$.
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Describe the sequence $9, 14, 19, 24, \text{...}$
Describe the sequence $9, 14, 19, 24, \text{...}$
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Arithmetic sequence. Each term increases by $5$ (constant difference).
Arithmetic sequence. Each term increases by $5$ (constant difference).
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Describe the sequence $7, 21, 63, \text{...}$
Describe the sequence $7, 21, 63, \text{...}$
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Geometric sequence. Each term is multiplied by $3$ (constant ratio).
Geometric sequence. Each term is multiplied by $3$ (constant ratio).
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Calculate the 5th term: $10, 30, 90, \text{...}$
Calculate the 5th term: $10, 30, 90, \text{...}$
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$810$. Common ratio is $3$, so $a_5 = 10 \times 3^{5-1} = 10 \times 81 = 810$.
$810$. Common ratio is $3$, so $a_5 = 10 \times 3^{5-1} = 10 \times 81 = 810$.
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Calculate the 7th term: $4, 8, 12, 16, \text{...}$
Calculate the 7th term: $4, 8, 12, 16, \text{...}$
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$28$. Common difference is $4$, so $a_7 = 4 + (7-1) \times 4 = 28$.
$28$. Common difference is $4$, so $a_7 = 4 + (7-1) \times 4 = 28$.
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Find the common ratio: $5, 15, 45, \text{...}$
Find the common ratio: $5, 15, 45, \text{...}$
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$3$. Divide consecutive terms: $15 \div 5 = 3$.
$3$. Divide consecutive terms: $15 \div 5 = 3$.
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If $a_1 = 5$ and $d = 3$, what is the $n$th term of the sequence?
If $a_1 = 5$ and $d = 3$, what is the $n$th term of the sequence?
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$a_n = 5 + (n-1) \times 3$. Substitute given values into the arithmetic sequence formula.
$a_n = 5 + (n-1) \times 3$. Substitute given values into the arithmetic sequence formula.
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If $a_1 = 2$ and $r = 4$, what is the $n$th term of the sequence?
If $a_1 = 2$ and $r = 4$, what is the $n$th term of the sequence?
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$a_n = 2 \times 4^{n-1}$. Substitute given values into the geometric sequence formula.
$a_n = 2 \times 4^{n-1}$. Substitute given values into the geometric sequence formula.
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Identify the sequence: $5, 9, 13, 17, \text{...}$
Identify the sequence: $5, 9, 13, 17, \text{...}$
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Arithmetic sequence. Each term adds $4$ to the previous term (constant difference).
Arithmetic sequence. Each term adds $4$ to the previous term (constant difference).
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Identify the sequence: $10, 20, 40, 80, \text{...}$
Identify the sequence: $10, 20, 40, 80, \text{...}$
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Geometric sequence. Each term doubles the previous term (constant ratio of $2$).
Geometric sequence. Each term doubles the previous term (constant ratio of $2$).
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