Algebra Flashcards: Arithmetic And Geometric Sequences As Functions

What this deck covers

This deck focuses on Arithmetic And Geometric Sequences As Functions, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra.

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: Model: A car value is \20{,}000$and keeps$0.85$of its value yearly. Write$V_n$ explicitly.

Answer: $V_n = 20000\left(0.85\right)^{n-1}$. Geometric sequence models exponential decay with factor $0.85$.

Flashcard 2: Find $a_6$ for the geometric sequence $a_1 = 1$ and $r = \frac{1}{3}$.

Answer: $a_6 = \frac{1}{243}$. Use $a_n = a_1 r^{n-1} = 1 \cdot \left(\frac{1}{3}\right)^5 = \frac{1}{243}$.

Flashcard 3: Write an explicit formula for the arithmetic sequence $5, 9, 13, 17, \dots$.

Answer: $a_n = 5 + 4(n-1)$. First term is $5$ and common difference is $4$.

Flashcard 4: What is the geometric mean of two positive numbers $x$ and $y$ (the middle term in a geometric sequence)?

Answer: $\sqrt{xy}$. The geometric mean is the square root of the product of two positive numbers.

Flashcard 5: What condition on consecutive differences identifies an arithmetic sequence?

Answer: All $a_n - a_{n-1}$ are equal. Constant differences between consecutive terms characterize arithmetic sequences.

Flashcard 6: Identify the sequence type if the terms change by adding a constant each time.

Answer: Arithmetic sequence. Adding a constant creates an arithmetic sequence with that common difference.

Flashcard 7: Find $a_6$ for the arithmetic sequence with $a_1 = 4$ and $d = 3$.

Answer: $a_6 = 19$. Use $a_n = a_1 + (n-1)d = 4 + 5(3) = 19$.

Flashcard 8: Identify the error: A student writes arithmetic explicit form as $a_n = a_1 + nd$. What is correct?

Answer: Correct: $a_n = a_1 + (n-1)d$. The exponent should be $(n-1)$, not $n$.

Flashcard 9: Given $a_3 = 11$ and $d = 4$ for an arithmetic sequence, find $a_1$.

Answer: $a_1 = 3$. Use $a_3 = a_1 + 2d$ to solve: $11 = a_1 + 8$.

Flashcard 10: Given $a_4 = 54$ and $r = 3$ for a geometric sequence, find $a_1$.

Answer: $a_1 = 2$. Use $a_4 = a_1 r^3$ to solve: $54 = a_1 \cdot 27$.

Flashcard 11: Find the next term in the geometric sequence $160, 80, 40, 20, \dots$.

Answer: $10$. Multiply the last term $20$ by the common ratio $r = \frac{1}{2}$.

Flashcard 12: Identify the sequence type if the terms change by multiplying by a constant each time.

Answer: Geometric sequence. Multiplying by a constant creates a geometric sequence with that common ratio.

Flashcard 13: Write a recursive formula for the arithmetic sequence defined by $a_n = -2n + 9$.

Answer: $a_1 = 7$; $a_n = a_{n-1} - 2$. Convert $a_n = -2n + 9$ to recursive by finding $a_1 = 7$ and $d = -2$.

Flashcard 14: Model: A car value is \20{,}000$and keeps$0.85$of its value yearly. Write$V_n$ explicitly.

Answer: $V_n = 20000\left(0.85\right)^{n-1}$. Geometric sequence models exponential decay with factor $0.85$.

Flashcard 15: Find $a_6$ for the arithmetic sequence with $a_1 = 4$ and $d = 3$.

Answer: $a_6 = 19$. Use $a_n = a_1 + (n-1)d = 4 + 5(3) = 19$.

Flashcard 16: Find the missing term to make $4, x, 16$ an arithmetic sequence.

Answer: $x = 10$. Use arithmetic mean: $x = \frac{4 + 16}{2} = 10$.

Flashcard 17: Model: A phone value is $900$ and loses $120$ each year. Write a recursive rule for $V_n$.

Answer: $V_1 = 900$; $V_n = V_{n-1} - 120$. Recursive form models constant yearly decreases.

Flashcard 18: Find $r$ for a geometric sequence with $a_2 = 6$ and $a_5 = 162$.

Answer: $r = 3$. Use $\frac{a_5}{a_2} = r^3$ to solve: $\frac{162}{6} = 27 = r^3$.

Flashcard 19: What is the recursive formula for an arithmetic sequence with first term $a_1$ and common difference $d$?

Answer: $a_1$ given; $a_n = a_{n-1} + d$. Each term equals the previous term plus the common difference $d$.

Flashcard 20: Given $a_1 = 6$ and $a_2 = 15$ for an arithmetic sequence, what is $d$?

Answer: $d = 9$. Common difference: $d = a_2 - a_1 = 15 - 6 = 9$.

Flashcard 21: What is the recursive formula for an arithmetic sequence with first term $a_1$ and common difference $d$?

Answer: $a_1$ given; $a_n = a_{n-1} + d$. Each term equals the previous term plus the common difference $d$.

Flashcard 22: Identify the error: A student writes arithmetic explicit form as $a_n = a_1 + nd$. What is correct?

Answer: Correct: $a_n = a_1 + (n-1)d$. The exponent should be $(n-1)$, not $n$.

Flashcard 23: Find the common difference $d$ for the arithmetic sequence $7, 3, -1, -5, \dots$.

Answer: $d = -4$. Subtract consecutive terms: $3 - 7 = -4$.

Flashcard 24: What is the recursive formula for a geometric sequence with first term $a_1$ and common ratio $r$?

Answer: $a_1$ given; $a_n = r a_{n-1}$. Each term equals the previous term multiplied by the common ratio $r$.

Flashcard 25: Write an explicit formula for the geometric sequence $81, 27, 9, 3, \dots$.

Answer: $a_n = 81\left(\frac{1}{3}\right)^{n-1}$. First term is $81$ and common ratio is $\frac{1}{3}$.

Flashcard 26: Find the next term in the arithmetic sequence $-4, -1, 2, 5, \dots$.

Answer: $8$. Add the common difference $d = 3$ to the last term $5$.

Flashcard 27: What is the common ratio $r$ in a geometric sequence in terms of consecutive terms $a_n$ and $a_{n-1}$?

Answer: $r = \frac{a_n}{a_{n-1}}$. The common ratio is found by dividing consecutive terms.

Flashcard 28: Translate $a_1 = 10$, $a_n = a_{n-1} - 2$ into an explicit formula for $a_n$.

Answer: $a_n = 10 - 2(n-1)$. Use $a_n = a_1 + (n-1)d$ with $d = -2$.

Flashcard 29: Identify the sequence type if the terms change by multiplying by a constant each time.

Answer: Geometric sequence. Multiplying by a constant creates a geometric sequence with that common ratio.

Flashcard 30: Find the missing term to make $4, x, 16$ an arithmetic sequence.

Answer: $x = 10$. Use arithmetic mean: $x = \frac{4 + 16}{2} = 10$.

Flashcard 31: Find the common difference $d$ for the arithmetic sequence $7, 3, -1, -5, \dots$.

Answer: $d = -4$. Subtract consecutive terms: $3 - 7 = -4$.

Flashcard 32: Find $a_1$ for the arithmetic sequence with $d = -7$ and $a_6 = 9$.

Answer: $a_1 = 44$. Use $a_6 = a_1 + 5d$ to solve: $9 = a_1 - 35$.

Flashcard 33: What condition on consecutive ratios identifies a geometric sequence?

Answer: All $\frac{a_n}{a_{n-1}}$ are equal. Constant ratios between consecutive terms characterize geometric sequences.

Flashcard 34: What is the explicit formula for an arithmetic sequence with first term $a_1$ and common difference $d$?

Answer: $a_n = a_1 + (n-1)d$. This formula adds $(n-1)$ multiples of the common difference $d$ to the first term.

Flashcard 35: Find the missing term to make $4, x, 16$ a geometric sequence with positive ratio.

Answer: $x = 8$. Use geometric mean: $x = \sqrt{4 \cdot 16} = 8$.

Flashcard 36: Find $d$ for an arithmetic sequence with $a_2 = 9$ and $a_5 = 21$.

Answer: $d = 4$. Use $a_5 - a_2 = 3d$ to solve: $21 - 9 = 12 = 3d$.

Flashcard 37: Given $a_n = 3\left(\frac{1}{2}\right)^{n-1}$, what are $a_1$ and $r$?

Answer: $a_1 = 3$, $r = \frac{1}{2}$. Identify first term and common ratio from explicit form.

Flashcard 38: Write $a_{n-1}$ in terms of $a_n$ and $r$ for a geometric sequence (assume $r \ne 0$).

Answer: $a_{n-1} = \frac{a_n}{r}$. Rearrange the geometric sequence relationship.

Flashcard 39: What is the common difference $d$ in an arithmetic sequence in terms of consecutive terms $a_n$ and $a_{n-1}$?

Answer: $d = a_n - a_{n-1}$. The common difference is found by subtracting consecutive terms.

Flashcard 40: Write a recursive formula for the arithmetic sequence defined by $a_n = -2n + 9$.

Answer: $a_1 = 7$; $a_n = a_{n-1} - 2$. Convert $a_n = -2n + 9$ to recursive by finding $a_1 = 7$ and $d = -2$.

Flashcard 41: Translate $a_n = 7 + 6(n-1)$ into a recursive formula.

Answer: $a_1 = 7$; $a_n = a_{n-1} + 6$. Convert explicit form to recursive by identifying $a_1$ and $d$.

Flashcard 42: Find the common ratio $r$ for the geometric sequence $2, 6, 18, 54, \dots$.

Answer: $r = 3$. Divide consecutive terms: $\frac{6}{2} = 3$.

Flashcard 43: Model: A population starts at $500$ and grows by a factor of $1.08$ yearly. What is $P_n$?

Answer: $P_n = 500\left(1.08\right)^{n-1}$. Geometric sequence models constant growth factor.

Flashcard 44: Find $a_8$ for the arithmetic sequence $a_1 = -5$ and $d = 2$.

Answer: $a_8 = 9$. Use $a_n = a_1 + (n-1)d = -5 + 7(2) = 9$.

Flashcard 45: Given $a_n = 3\left(\frac{1}{2}\right)^{n-1}$, what are $a_1$ and $r$?

Answer: $a_1 = 3$, $r = \frac{1}{2}$. Identify first term and common ratio from explicit form.

Flashcard 46: What is the common difference $d$ in an arithmetic sequence in terms of consecutive terms $a_n$ and $a_{n-1}$?

Answer: $d = a_n - a_{n-1}$. The common difference is found by subtracting consecutive terms.

Flashcard 47: What is the geometric mean of two positive numbers $x$ and $y$ (the middle term in a geometric sequence)?

Answer: $\sqrt{xy}$. The geometric mean is the square root of the product of two positive numbers.

Flashcard 48: Identify whether $-1, 2, 5, 8, \dots$ is arithmetic or geometric.

Answer: Arithmetic. Differences are constant: $2-(-1) = 3$.

Flashcard 49: Translate $a_1 = 5$, $a_n = 3a_{n-1}$ into an explicit formula for $a_n$.

Answer: $a_n = 5\cdot 3^{n-1}$. Use $a_n = a_1 r^{n-1}$ with $r = 3$.

Flashcard 50: Write $a_{n-1}$ in terms of $a_n$ and $r$ for a geometric sequence (assume $r \ne 0$).

Answer: $a_{n-1} = \frac{a_n}{r}$. Rearrange the geometric sequence relationship.

Flashcard 51: Write a recursive formula for the arithmetic sequence $5, 9, 13, 17, \dots$.

Answer: $a_1 = 5$; $a_n = a_{n-1} + 4$. First term is $5$ and each term adds $4$ to the previous.

Flashcard 52: Given $a_n = 12 + 5(n-1)$, what are $a_1$ and $d$?

Answer: $a_1 = 12$, $d = 5$. Identify first term and common difference from explicit form.

Flashcard 53: Identify the error: A student writes geometric explicit form as $a_n = a_1 r^n$. What is correct?

Answer: Correct: $a_n = a_1 r^{n-1}$. The exponent should be $(n-1)$, not $n$.

Flashcard 54: Find $r$ for a geometric sequence with $a_2 = 6$ and $a_5 = 162$.

Answer: $r = 3$. Use $\frac{a_5}{a_2} = r^3$ to solve: $\frac{162}{6} = 27 = r^3$.

Flashcard 55: Given $a_1 = 6$ and $a_2 = 15$ for a geometric sequence, what is $r$?

Answer: $r = \frac{5}{2}$. Common ratio: $r = \frac{a_2}{a_1} = \frac{15}{6} = \frac{5}{2}$.

Flashcard 56: Model: A salary starts at \40{,}000$and increases by$$1{,}500$yearly. What is$S_n$?

Answer: $S_n = 40000 + 1500(n-1)$. Arithmetic sequence models constant yearly increases.

Flashcard 57: Write an explicit formula for the arithmetic sequence $5, 9, 13, 17, \dots$.

Answer: $a_n = 5 + 4(n-1)$. First term is $5$ and common difference is $4$.

Flashcard 58: Translate $a_n = 9\left(-2\right)^{n-1}$ into a recursive formula.

Answer: $a_1 = 9$; $a_n = -2a_{n-1}$. Convert explicit form to recursive by identifying $a_1$ and $r$.

Flashcard 59: Translate $a_1 = 10$, $a_n = a_{n-1} - 2$ into an explicit formula for $a_n$.

Answer: $a_n = 10 - 2(n-1)$. Use $a_n = a_1 + (n-1)d$ with $d = -2$.

Flashcard 60: Translate $a_n = 7 + 6(n-1)$ into a recursive formula.

Answer: $a_1 = 7$; $a_n = a_{n-1} + 6$. Convert explicit form to recursive by identifying $a_1$ and $d$.

Flashcard 61: Find $d$ for an arithmetic sequence with $a_2 = 9$ and $a_5 = 21$.

Answer: $d = 4$. Use $a_5 - a_2 = 3d$ to solve: $21 - 9 = 12 = 3d$.

Flashcard 62: Write a recursive formula for the geometric sequence $81, 27, 9, 3, \dots$.

Answer: $a_1 = 81$; $a_n = \frac{1}{3}a_{n-1}$. First term is $81$ and each term is $\frac{1}{3}$ times the previous.

Flashcard 63: Write $a_{n-1}$ in terms of $a_n$ and $d$ for an arithmetic sequence.

Answer: $a_{n-1} = a_n - d$. Rearrange the arithmetic sequence relationship.

Flashcard 64: Write an explicit formula for the geometric sequence with $a_1 = -3$ and $r = -2$.

Answer: $a_n = -3\left(-2\right)^{n-1}$. Use explicit form $a_n = a_1 r^{n-1}$ with given values.

Flashcard 65: Identify whether $-1, 2, 5, 8, \dots$ is arithmetic or geometric.

Answer: Arithmetic. Differences are constant: $2-(-1) = 3$.

Flashcard 66: Given $a_1 = 6$ and $a_2 = 15$ for a geometric sequence, what is $r$?

Answer: $r = \frac{5}{2}$. Common ratio: $r = \frac{a_2}{a_1} = \frac{15}{6} = \frac{5}{2}$.

Flashcard 67: Write an explicit formula for the geometric sequence $81, 27, 9, 3, \dots$.

Answer: $a_n = 81\left(\frac{1}{3}\right)^{n-1}$. First term is $81$ and common ratio is $\frac{1}{3}$.

Flashcard 68: What is the recursive formula for a geometric sequence with first term $a_1$ and common ratio $r$?

Answer: $a_1$ given; $a_n = r a_{n-1}$. Each term equals the previous term multiplied by the common ratio $r$.

Flashcard 69: Identify the sequence type if the terms change by adding a constant each time.

Answer: Arithmetic sequence. Adding a constant creates an arithmetic sequence with that common difference.

Flashcard 70: Find $a_8$ for the arithmetic sequence $a_1 = -5$ and $d = 2$.

Answer: $a_8 = 9$. Use $a_n = a_1 + (n-1)d = -5 + 7(2) = 9$.

Flashcard 71: What is the arithmetic mean of two numbers $x$ and $y$ (the middle term in an arithmetic sequence)?

Answer: $\frac{x+y}{2}$. The arithmetic mean is the average of two numbers.

Flashcard 72: Identify whether $3, -6, 12, -24, \dots$ is arithmetic or geometric.

Answer: Geometric. Ratios are constant: $\frac{-6}{3} = -2$.

Flashcard 73: Given $a_4 = 54$ and $r = 3$ for a geometric sequence, find $a_1$.

Answer: $a_1 = 2$. Use $a_4 = a_1 r^3$ to solve: $54 = a_1 \cdot 27$.

Flashcard 74: Find $a_1$ for the geometric sequence with $r = 2$ and $a_6 = 96$.

Answer: $a_1 = 3$. Use $a_6 = a_1 r^5$ to solve: $96 = a_1 \cdot 32$.

Flashcard 75: Model: A salary starts at \40{,}000 $and increases by$ $1{,}500 $yearly. What is$S_n$?

Answer: $S_n = 40000 + 1500(n-1)$. Arithmetic sequence models constant yearly increases.

Flashcard 76: Write $a_{n-1}$ in terms of $a_n$ and $d$ for an arithmetic sequence.

Answer: $a_{n-1} = a_n - d$. Rearrange the arithmetic sequence relationship.

Flashcard 77: Find $a_1$ for the geometric sequence with $r = 2$ and $a_6 = 96$.

Answer: $a_1 = 3$. Use $a_6 = a_1 r^5$ to solve: $96 = a_1 \cdot 32$.

Flashcard 78: Write a recursive formula for the geometric sequence $81, 27, 9, 3, \dots$.

Answer: $a_1 = 81$; $a_n = \frac{1}{3}a_{n-1}$. First term is $81$ and each term is $\frac{1}{3}$ times the previous.

Flashcard 79: Find the next term in the arithmetic sequence $-4, -1, 2, 5, \dots$.

Answer: $8$. Add the common difference $d = 3$ to the last term $5$.

Flashcard 80: What condition on consecutive differences identifies an arithmetic sequence?

Answer: All $a_n - a_{n-1}$ are equal. Constant differences between consecutive terms characterize arithmetic sequences.

Flashcard 81: Write a recursive formula for the arithmetic sequence $5, 9, 13, 17, \dots$.

Answer: $a_1 = 5$; $a_n = a_{n-1} + 4$. First term is $5$ and each term adds $4$ to the previous.

Flashcard 82: What is the common ratio $r$ in a geometric sequence in terms of consecutive terms $a_n$ and $a_{n-1}$?

Answer: $r = \frac{a_n}{a_{n-1}}$. The common ratio is found by dividing consecutive terms.

Flashcard 83: Find the missing term to make $4, x, 16$ a geometric sequence with positive ratio.

Answer: $x = 8$. Use geometric mean: $x = \sqrt{4 \cdot 16} = 8$.

Flashcard 84: What is the explicit formula for a geometric sequence with first term $a_1$ and common ratio $r$?

Answer: $a_n = a_1 r^{n-1}$. This formula multiplies the first term by $r$ raised to the $(n-1)$th power.

Flashcard 85: Find $a_6$ for the geometric sequence $a_1 = 1$ and $r = \frac{1}{3}$.

Answer: $a_6 = \frac{1}{243}$. Use $a_n = a_1 r^{n-1} = 1 \cdot \left(\frac{1}{3}\right)^5 = \frac{1}{243}$.

Flashcard 86: What is the arithmetic mean of two numbers $x$ and $y$ (the middle term in an arithmetic sequence)?

Answer: $\frac{x+y}{2}$. The arithmetic mean is the average of two numbers.

Flashcard 87: Model: A phone value is $900 and loses $120 each year. Write a recursive rule for $V_n$.

Answer: $V_1 = 900$; $V_n = V_{n-1} - 120$. Recursive form models constant yearly decreases.

Flashcard 88: Find the next term in the geometric sequence $160, 80, 40, 20, \dots$.

Answer: $10$. Multiply the last term $20$ by the common ratio $r = \frac{1}{2}$.

Flashcard 89: Write an explicit formula for the geometric sequence with $a_1 = -3$ and $r = -2$.

Answer: $a_n = -3\left(-2\right)^{n-1}$. Use explicit form $a_n = a_1 r^{n-1}$ with given values.

Flashcard 90: Given $a_n = 12 + 5(n-1)$, what are $a_1$ and $d$?

Answer: $a_1 = 12$, $d = 5$. Identify first term and common difference from explicit form.

Flashcard 91: Model: A population starts at $500$ and grows by a factor of $1.08$ yearly. What is $P_n$?

Answer: $P_n = 500\left(1.08\right)^{n-1}$. Geometric sequence models constant growth factor.

Flashcard 92: What condition on consecutive ratios identifies a geometric sequence?

Answer: All $\frac{a_n}{a_{n-1}}$ are equal. Constant ratios between consecutive terms characterize geometric sequences.

Flashcard 93: Find $a_1$ for the arithmetic sequence with $d = -7$ and $a_6 = 9$.

Answer: $a_1 = 44$. Use $a_6 = a_1 + 5d$ to solve: $9 = a_1 - 35$.