Algebra 2 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 2.

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: What is the formula for the common difference $d$ using two terms $a_k$ and $a_m$ of an arithmetic sequence?

Answer: $d=\frac{a_m-a_k}{m-k}$. Difference formula: change in terms divided by change in positions.

Flashcard 2: Find $a_7$ for the arithmetic sequence $a_n=2+5(n-1)$.

Answer: $a_7=32$. Using $a_n=2+5(n-1)$: $a_7=2+5(6)=2+30=32$.

Flashcard 3: Find $a_1$ for a geometric sequence with $a_3=36$ and $r=3$.

Answer: $a_1=4$. Working backwards: $a_1=\frac{a_3}{r^2}=\frac{36}{3^2}=\frac{36}{9}=4$.

Flashcard 4: Find $a_1$ for a geometric sequence with $a_3=36$ and $r=3$.

Answer: $a_1=4$. Working backwards: $a_1=\frac{a_3}{r^2}=\frac{36}{3^2}=\frac{36}{9}=4$.

Flashcard 5: Translate to explicit form: $a_1=7$, $a_n=a_{n-1}-2$ for $n\ge^2$.

Answer: $a_n=7+(n-1)(-2)$. Converting recursive to explicit: $d=-2$, so $a_n=7+(n-1)(-2)$.

Flashcard 6: Translate to explicit form: $a_1=5$, $a_n=\frac{1}{2}a_{n-1}$ for $n\ge^2$.

Answer: $a_n=5\left(\frac{1}{2}\right)^{n-1}$. Converting recursive to explicit: $r=\frac{1}{2}$, so $a_n=5\left(\frac{1}{2}\right)^{n-1}$.

Flashcard 7: What is the explicit formula for the arithmetic sequence with $a_1=12$ and $d=-3$?

Answer: $a_n=12+(n-1)(-3)$. Uses the standard explicit formula with $a_1=12$ and $d=-3$.

Flashcard 8: What is the explicit formula for the geometric sequence with $a_1=2$ and $r=\frac{1}{4}$?

Answer: $a_n=2\left(\frac{1}{4}\right)^{n-1}$. Uses the standard explicit formula with $a_1=2$ and $r=\frac{1}{4}$.

Flashcard 9: Identify the arithmetic sequence: $3,7,11,15,\dots$; what is the common difference $d$?

Answer: $d=4$. Each term increases by $4$: $7-3=4$, $11-7=4$, etc.

Flashcard 10: What is the recursive definition for the arithmetic sequence with $a_1=-5$ and $d=6$?

Answer: $a_1=-5$; $a_n=a_{n-1}+6$. Standard recursive form with given first term and difference.

Flashcard 11: Translate to recursive form: $a_n=3+4(n-1)$ with $n\ge^1$.

Answer: $a_1=3$; $a_n=a_{n-1}+4$. Converting explicit to recursive: $a_1=3$ and $d=4$.

Flashcard 12: Translate to recursive form: $a_n=8\cdot(-3)^{n-1}$ with $n\ge^1$.

Answer: $a_1=8$; $a_n=-3\,a_{n-1}$. Converting explicit to recursive: $a_1=8$ and $r=-3$.

Flashcard 13: Find and correct the error: Geometric explicit written as $a_n=a_1\cdot r^{n}$.

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

Flashcard 14: Which option is the common difference for the arithmetic sequence $-2,1,4,7,\dots$?

Answer: $d=3$. Each term increases by the same amount: $1-(-2)=3$.

Flashcard 15: Which option is the common ratio for the geometric sequence $-4,12,-36,108,\dots$?

Answer: $r=-3$. Each term is multiplied by $-3$: $12÷(-4)=-3$.

Flashcard 16: Identify whether $a_n=2n+1$ defines an arithmetic or geometric sequence.

Answer: Arithmetic. Linear form $a_n=2n+1$ has constant differences between consecutive terms.

Flashcard 17: Identify whether $a_n=7\cdot 5^{n}$ defines an arithmetic or geometric sequence.

Answer: Geometric. Exponential form $a_n=7\cdot 5^n$ has constant ratios between consecutive terms.

Flashcard 18: What is the geometric mean formula that gives the middle positive term between $a$ and $b$ in a geometric sequence?

Answer: $\sqrt{ab}$. Square root of the product of two consecutive positive terms.

Flashcard 19: Find $d$ if an arithmetic sequence has $a_n=18-6(n-1)$.

Answer: $d=-6$. The coefficient of $(n-1)$ gives the common difference directly.

Flashcard 20: Find $r$ if a geometric sequence has $a_n=10\cdot\left(\frac{3}{2}\right)^{n-1}$.

Answer: $r=\frac{3}{2}$. The base of the exponent gives the common ratio directly.

Flashcard 21: Find $a_7$ for the arithmetic sequence $a_n=2+5(n-1)$.

Answer: $a_7=32$. Using $a_n=2+5(n-1)$: $a_7=2+5(6)=2+30=32$.

Flashcard 22: Find $a_6$ for the geometric sequence $a_n=\left(-2\right)\cdot 3^{n-1}$.

Answer: $a_6=-486$. Using $a_n=(-2)\cdot 3^{n-1}$: $a_6=(-2)\cdot 3^5=(-2)\cdot 243=-486$.

Flashcard 23: Find the arithmetic mean between $8$ and $20$.

Answer: $14$. Using $\frac{a+b}{2}$: $\frac{8+20}{2}=\frac{28}{2}=14$.

Flashcard 24: Find the geometric mean between $4$ and $36$ (assume positive).

Answer: $12$. Using $\sqrt{ab}$: $\sqrt{4\cdot 36}=\sqrt{144}=12$.

Flashcard 25: What is the explicit formula for an arithmetic sequence written from term $a_k$ instead of $a_1$?

Answer: $a_n=a_k+(n-k)d$. General explicit form starting from any known term $a_k$.

Flashcard 26: What is the formula for the common ratio $r$ using two terms $a_k$ and $a_m$ of a geometric sequence?

Answer: $r=\left(\frac{a_m}{a_k}\right)^{\frac{1}{m-k}}$. Ratio formula using the $(m-k)$th root of the term quotient.

Flashcard 27: What is the explicit formula for a geometric sequence written from term $a_k$ instead of $a_1$?

Answer: $a_n=a_k\cdot r^{n-k}$. General explicit form starting from any known term $a_k$.

Flashcard 28: Use $a_n=a_k+(n-k)d$: If $a_4=9$ and $d=2$, what is $a_{10}$?

Answer: $a_{10}=21$. Using the general form: $a_{10}=9+(10-4)(2)=9+12=21$.

Flashcard 29: Use $a_n=a_k\cdot r^{n-k}$: If $a_3=16$ and $r=\frac{1}{2}$, what is $a_7$?

Answer: $a_7=1$. Using the general form: $a_7=16\cdot\left(\frac{1}{2}\right)^{7-3}=16\cdot\frac{1}{16}=1$.

Flashcard 30: Find and correct the error: Arithmetic explicit written as $a_n=a_1+nd$.

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