Algebra Flashcards: Deriving Applying The Geometric Series Formula

What this deck covers

This deck focuses on Deriving Applying The Geometric Series Formula, 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: Convert an annual interest rate $R$ compounded monthly to periodic rate $i$. What is $i$?

Answer: $i=\frac{R}{12}$. Divide annual rate by 12 for monthly compounding.

Flashcard 2: Find $S_5$ for $a_1=-3$ and $r=-2$.

Answer: $S_5=-3\frac{1-(-2)^5}{1-(-2)}=-33$. Apply formula with $a_1=-3$, $r=-2$, $n=5$.

Flashcard 3: What is the sum of the first $n$ terms if $r=1$ and first term is $a_1$?

Answer: $S_n=na_1$. When $r=1$, all terms equal $a_1$, so sum is $n$ times $a_1$.

Flashcard 4: Identify the number of terms $n$ in the geometric series $5+15+45+135$.

Answer: $n=4$. Count the terms: $5, 15, 45, 135$ gives $n=4$.

Flashcard 5: Compute $n$ for a $15$-year loan with monthly payments.

Answer: $n=180$. Calculate $n=15\times 12=180$ monthly payments.

Flashcard 6: Identify the common ratio $r$ of the geometric sequence $7,21,63,\cdots$.

Answer: $r=3$. Divide consecutive terms: $\frac{21}{7}=3$ and $\frac{63}{21}=3$.

Flashcard 7: Find the sum $S_3$ for $a_1=1$ and $r=10$ using the finite sum formula.

Answer: $S_3=1\frac{1-10^3}{1-10}=111$. Apply geometric sum formula with $a_1=1$ and $r=10$.

Flashcard 8: For a loan, what is the common ratio $r$ in the present value geometric series of discounted payments?

Answer: $r=\frac{1}{1+i}$. Each payment is discounted by factor $(1+i)^{-1}$.

Flashcard 9: Find $a_1$ if $S_3=21$, $r=2$, and $n=3$.

Answer: $a_1=3$. Use $S_3=a_1\frac{1-r^3}{1-r}=a_1\frac{1-8}{1-2}=21$ to solve.

Flashcard 10: Compute $S_4$ for $a_1=\frac{1}{3}$ and $r=3$.

Answer: $S_4=\frac{1}{3}\frac{1-3^4}{1-3}=\frac{40}{3}$. Apply formula with $a_1=\frac{1}{3}$, $r=3$, $n=4$.

Flashcard 11: What is the first term $a_1$ in the discounted-payment series $\text{PMT}(1+i)^{-1}+\cdots$?

Answer: $a_1=\text{PMT}(1+i)^{-1}$. First payment discounted by one period.

Flashcard 12: Identify the common ratio $r$ for the series $-3,6,-12,\cdots$.

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

Flashcard 13: Compute $\text{PMT}$ if $PV=1000$, $i=0.25$, and $n=1$.

Answer: $\text{PMT}=1000\frac{0.25}{1-(1.25)^{-1}}=1250$. Solve payment formula: $1000\times\frac{0.25}{1-0.8}=1250$.

Flashcard 14: Compute $S_3$ for $a_1=6$ and $r=-\frac{1}{2}$.

Answer: $S_3=6\frac{1-(-\frac{1}{2})^3}{1+\frac{1}{2}}=\frac{9}{2}$. Apply formula with $a_1=6$, $r=-\frac{1}{2}$, and $n=3$.

Flashcard 15: Find $n$ if $a_1=3$, $r=2$, and $a_n=96$.

Answer: $n=6$. Use $a_n=a_1r^{n-1}$ to solve: $96=3\cdot 2^{n-1}$.

Flashcard 16: Find $a_5$ for a geometric sequence with $a_1=2$ and $r=3$.

Answer: $a_5=2\cdot 3^4=162$. Use $a_n=a_1r^{n-1}$ with $n=5$, $a_1=2$, $r=3$.

Flashcard 17: What is the common ratio $r$ in a geometric sequence in terms of consecutive terms?

Answer: $r=\frac{a_{n+1}}{a_n}$. The ratio of any term to the previous term gives the common ratio.

Flashcard 18: Find and correct the missing condition in $S_n=a_1\frac{1-r^n}{1-r}$: when is it valid?

Answer: Valid only when $r\ne 1$. Formula undefined when $r=1$ due to division by zero.

Flashcard 19: Find the sum $S_3$ for $a_1=1$ and $r=-1$.

Answer: $S_3=1\frac{1-(-1)^3}{1-(-1)}=1$. Apply formula: $1\frac{1-(-1)^3}{1-(-1)}=1\frac{2}{2}=1$.

Flashcard 20: State the formula for the finite geometric sum $S_n=a_1+a_1r+\cdots+a_1r^{n-1}$ for $r\ne 1$.

Answer: $S_n=a_1\frac{1-r^n}{1-r}$. Derived by multiplying by $r$ and subtracting to eliminate middle terms.

Flashcard 21: What is the last term $a_n$ in a geometric series expressed using $a_1$, $r$, and $n$?

Answer: $a_n=a_1r^{n-1}$. The final term uses the same geometric sequence formula.

Flashcard 22: What is the $n$th term $a_n$ in terms of $a_1$ and $r$ for a geometric sequence?

Answer: $a_n=a_1r^{n-1}$. General term formula where position determines the power of $r$.

Flashcard 23: Find $S_5$ for $a_1=-3$ and $r=-2$.

Answer: $S_5=-3\frac{1-(-2)^5}{1-(-2)}=-33$. Apply formula with $a_1=-3$, $r=-2$, $n=5$.

Flashcard 24: Identify the first term $a_1$ of the geometric series $7+21+63+\cdots$.

Answer: $a_1=7$. The first term is the initial value in the sequence.

Flashcard 25: Find $PV$ if $\text{PMT}=200$, $i=0.01$, and $n=2$ for an ordinary annuity.

Answer: $PV=200\frac{1-(1.01)^{-2}}{0.01}$. Apply present value formula with given payment and rate.

Flashcard 26: Find $a_1$ if $a_3=20$ and $r=2$ for a geometric sequence.

Answer: $a_1=\frac{20}{2^2}=5$. Use $a_3=a_1r^2$ to solve: $20=a_1\cdot 2^2$.

Flashcard 27: Find $S_2$ for $a_1=8$ and $r=\frac{1}{4}$.

Answer: $S_2=8\frac{1-(\frac{1}{4})^2}{1-\frac{1}{4}}=10$. Apply formula with $a_1=8$, $r=\frac{1}{4}$, $n=2$.

Flashcard 28: What is the definition of a geometric sequence using first term $a_1$ and ratio $r$?

Answer: $a_n=a_1r^{n-1}$. Each term is the first term multiplied by $r$ raised to the term position minus 1.

Flashcard 29: Compute $S_4$ for the geometric series $2-1+\frac{1}{2}-\frac{1}{4}$.

Answer: $S_4=\frac{5}{4}$. Geometric series with $a_1=2$, $r=-\frac{1}{2}$, $n=4$.

Flashcard 30: Compute $S_5$ for $a_1=1$ and $r=\frac{1}{2}$.

Answer: $S_5=1\frac{1-(\frac{1}{2})^5}{1-\frac{1}{2}}=\frac{31}{16}$. Apply formula with $a_1=1$, $r=\frac{1}{2}$, and $n=5$.

Flashcard 31: What is the key multiplication step used to derive the geometric sum (multiply $S_n$ by what)?

Answer: Multiply by $r$ to form $rS_n$. Creates alignment to subtract and cancel telescoping middle terms.

Flashcard 32: What is the last term $a_n$ in a geometric series expressed using $a_1$, $r$, and $n$?

Answer: $a_n=a_1r^{n-1}$. The final term uses the same geometric sequence formula.

Flashcard 33: If a loan has term $t$ years with monthly payments, what is the number of payments $n$?

Answer: $n=12t$. Multiply years by 12 payments per year.

Flashcard 34: Compute the sum $S_4$ for the series $5+15+45+135$.

Answer: $S_4=5\frac{1-3^4}{1-3}=200$. Apply formula with $a_1=5$, $r=3$, $n=4$.

Flashcard 35: Find $r$ if $a_1=2$, $a_4=54$, and the sequence is geometric.

Answer: $r=3$. Use $a_4=a_1r^3$ to solve: $54=2r^3$, so $r^3=27$.

Flashcard 36: Identify the error: using $S_n=a_1\frac{1-r^n}{1-r}$ when $r=1$. What is the correct sum?

Answer: Use $S_n=na_1$ when $r=1$. When $r=1$, the geometric sum formula has zero denominator.

Flashcard 37: Compute the monthly rate $i$ for an APR of $6\%$ compounded monthly.

Answer: $i=\frac{0.06}{12}=0.005$. Divide APR by 12: $\frac{6\%}{12}=0.5\%=0.005$.

Flashcard 38: Identify the geometric series that represents $PV$ of $n$ payments $\text{PMT}$ at rate $i$ (ordinary annuity).

Answer: $PV=\text{PMT}\sum_{k=1}^{n}(1+i)^{-k}$. Sum of discounted payments using geometric series.

Flashcard 39: In the discounted-payment series, what is the $n$th term $a_n$?

Answer: $a_n=\text{PMT}(1+i)^{-n}$. Last payment discounted by $n$ periods.

Flashcard 40: Compute $S_4$ for $a_1=3$ and $r=-2$.

Answer: $S_4=3\frac{1-(-2)^4}{1-(-2)}=-15$. Use formula with negative ratio $r=-2$ and $n=4$.

Flashcard 41: Find the sum $S_3$ for $a_1=1$ and $r=10$ using the finite sum formula.

Answer: $S_3=1 \frac{1-10^3}{1-10}=111$. Apply geometric sum formula with $a_1=1$ and $r=10$.

Flashcard 42: Compute $S_3$ for the geometric series $10+5+2.5$.

Answer: $S_3=17.5$. Direct addition: $10+5+2.5=17.5$.

Flashcard 43: Compute $S_2$ for $a_1=9$ and $r=\frac{2}{3}$.

Answer: $S_2=9\frac{1-(\frac{2}{3})^2}{1-\frac{2}{3}}=15$. Apply formula with $a_1=9$, $r=\frac{2}{3}$, and $n=2$.

Flashcard 44: Compute $S_4$ for $a_1=5$, $r=2$ using the geometric sum formula.

Answer: $S_4=5\frac{1-2^4}{1-2}=75$. Apply the geometric sum formula with $n=4$ terms.

Flashcard 45: Compute the common ratio $r$ for discounted payments when $i=0.05$ per period.

Answer: $r=\frac{1}{1.05}$. Common ratio is the discount factor $\frac{1}{1+i}$.

Flashcard 46: What condition must hold to divide by $(1-r)$ when deriving the sum formula?

Answer: $r\ne 1$. Division by zero occurs if $r=1$, making the formula undefined.

Flashcard 47: Identify the first term $a_1$ of the geometric series $7+21+63+\cdots$.

Answer: $a_1=7$. The first term is the initial value in the sequence.

Flashcard 48: What is the present value formula for an ordinary annuity with payment $\text{PMT}$, rate $i$, and $n$ payments?

Answer: $PV=\text{PMT}\frac{1-(1+i)^{-n}}{i}$. Geometric series formula for discounted future payments.

Flashcard 49: In the derivation, what expression results from subtracting: $S_n-rS_n$?

Answer: $(1-r)S_n=a_1-a_1r^n$. Factor out $(1-r)$ from the left side after subtraction.

Flashcard 50: Find the common ratio $r$ for the sequence $81,27,9,\cdots$.

Answer: $r=\frac{1}{3}$. Divide consecutive terms: $\frac{27}{81}=\frac{1}{3}$.

Flashcard 51: What is the loan payment formula for principal $PV$, periodic rate $i$, and $n$ payments?

Answer: $\text{PMT}=PV\frac{i}{1-(1+i)^{-n}}$. Solve the present value formula for the payment amount.

Flashcard 52: Compute $S_6$ for $a_1=4$ and $r=\frac{1}{2}$.

Answer: $S_6=4\frac{1-(\frac{1}{2})^6}{1-\frac{1}{2}}=\frac{63}{8}$. Apply formula with $a_1=4$, $r=\frac{1}{2}$, and $n=6$.

Flashcard 53: State an equivalent finite geometric sum formula for $S_n$ using denominator $(r-1)$ for $r\ne 1$.

Answer: $S_n=a_1\frac{r^n-1}{r-1}$. Multiply numerator and denominator by $-1$ to get equivalent form.

Flashcard 54: What is the definition of a geometric sequence using first term $a_1$ and ratio $r$?

Answer: $a_n=a_1r^{n-1}$. Each term is the first term multiplied by $r$ raised to the term position minus 1.

Flashcard 55: Compute $PV$ if $\text{PMT}=100$, $i=0.25$, and $n=1$.

Answer: $PV=100\frac{1-(1.25)^{-1}}{0.25}=80$. Apply present value formula: $\text{PMT}\frac{1-1.25^{-1}}{0.25}=80$.

Flashcard 56: Find $r$ if $S_2=12$, $a_1=3$, and the series is geometric with $n=2$.

Answer: $r=3$. Use $S_2=a_1(1+r)=3(1+r)=12$ to solve for $r$.

Flashcard 57: Compute $S_3$ for the geometric series $10+5+2.5$.

Answer: $S_3=17.5$. Direct addition: $10+5+2.5=17.5$.