Deriving/Applying the Geometric Series Formula - Algebra
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Find $a_1$ if $S_3=21$, $r=2$, and $n=3$.
Find $a_1$ if $S_3=21$, $r=2$, and $n=3$.
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$a_1=3$. Use $S_3=a_1\frac{1-r^3}{1-r}=a_1\frac{1-8}{1-2}=21$ to solve.
$a_1=3$. Use $S_3=a_1\frac{1-r^3}{1-r}=a_1\frac{1-8}{1-2}=21$ to solve.
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What is the present value formula for an ordinary annuity with payment $\text{PMT}$, rate $i$, and $n$ payments?
What is the present value formula for an ordinary annuity with payment $\text{PMT}$, rate $i$, and $n$ payments?
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$PV=\text{PMT}\frac{1-(1+i)^{-n}}{i}$. Geometric series formula for discounted future payments.
$PV=\text{PMT}\frac{1-(1+i)^{-n}}{i}$. Geometric series formula for discounted future payments.
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What is the loan payment formula for principal $PV$, periodic rate $i$, and $n$ payments?
What is the loan payment formula for principal $PV$, periodic rate $i$, and $n$ payments?
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$\text{PMT}=PV\frac{i}{1-(1+i)^{-n}}$. Solve the present value formula for the payment amount.
$\text{PMT}=PV\frac{i}{1-(1+i)^{-n}}$. Solve the present value formula for the payment amount.
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For a loan, what is the common ratio $r$ in the present value geometric series of discounted payments?
For a loan, what is the common ratio $r$ in the present value geometric series of discounted payments?
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$r=\frac{1}{1+i}$. Each payment is discounted by factor $(1+i)^{-1}$.
$r=\frac{1}{1+i}$. Each payment is discounted by factor $(1+i)^{-1}$.
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Convert an annual interest rate $R$ compounded monthly to periodic rate $i$. What is $i$?
Convert an annual interest rate $R$ compounded monthly to periodic rate $i$. What is $i$?
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$i=\frac{R}{12}$. Divide annual rate by 12 for monthly compounding.
$i=\frac{R}{12}$. Divide annual rate by 12 for monthly compounding.
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Compute $n$ for a $15$-year loan with monthly payments.
Compute $n$ for a $15$-year loan with monthly payments.
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$n=180$. Calculate $n=15\times 12=180$ monthly payments.
$n=180$. Calculate $n=15\times 12=180$ monthly payments.
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Compute the monthly rate $i$ for an APR of $6%$ compounded monthly.
Compute the monthly rate $i$ for an APR of $6%$ compounded monthly.
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$i=\frac{0.06}{12}=0.005$. Divide APR by 12: $\frac{6%}{12}=0.5%=0.005$.
$i=\frac{0.06}{12}=0.005$. Divide APR by 12: $\frac{6%}{12}=0.5%=0.005$.
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Find $PV$ if $\text{PMT}=200$, $i=0.01$, and $n=2$ for an ordinary annuity.
Find $PV$ if $\text{PMT}=200$, $i=0.01$, and $n=2$ for an ordinary annuity.
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$PV=200\frac{1-(1.01)^{-2}}{0.01}$. Apply present value formula with given payment and rate.
$PV=200\frac{1-(1.01)^{-2}}{0.01}$. Apply present value formula with given payment and rate.
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What is the last term $a_n$ in a geometric series expressed using $a_1$, $r$, and $n$?
What is the last term $a_n$ in a geometric series expressed using $a_1$, $r$, and $n$?
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$a_n=a_1r^{n-1}$. The final term uses the same geometric sequence formula.
$a_n=a_1r^{n-1}$. The final term uses the same geometric sequence formula.
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Compute $S_4$ for $a_1=5$, $r=2$ using the geometric sum formula.
Compute $S_4$ for $a_1=5$, $r=2$ using the geometric sum formula.
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$S_4=5\frac{1-2^4}{1-2}=75$. Apply the geometric sum formula with $n=4$ terms.
$S_4=5\frac{1-2^4}{1-2}=75$. Apply the geometric sum formula with $n=4$ terms.
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Compute $S_3$ for the geometric series $10+5+2.5$.
Compute $S_3$ for the geometric series $10+5+2.5$.
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$S_3=17.5$. Direct addition: $10+5+2.5=17.5$.
$S_3=17.5$. Direct addition: $10+5+2.5=17.5$.
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Find $r$ if $S_2=12$, $a_1=3$, and the series is geometric with $n=2$.
Find $r$ if $S_2=12$, $a_1=3$, and the series is geometric with $n=2$.
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$r=3$. Use $S_2=a_1(1+r)=3(1+r)=12$ to solve for $r$.
$r=3$. Use $S_2=a_1(1+r)=3(1+r)=12$ to solve for $r$.
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Compute $S_5$ for $a_1=1$ and $r=\frac{1}{2}$.
Compute $S_5$ for $a_1=1$ and $r=\frac{1}{2}$.
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$S_5=1\frac{1-(\frac{1}{2})^5}{1-\frac{1}{2}}=\frac{31}{16}$. Apply formula with $a_1=1$, $r=\frac{1}{2}$, and $n=5$.
$S_5=1\frac{1-(\frac{1}{2})^5}{1-\frac{1}{2}}=\frac{31}{16}$. Apply formula with $a_1=1$, $r=\frac{1}{2}$, and $n=5$.
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Identify the first term $a_1$ of the geometric series $7+21+63+\cdots$.
Identify the first term $a_1$ of the geometric series $7+21+63+\cdots$.
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$a_1=7$. The first term is the initial value in the sequence.
$a_1=7$. The first term is the initial value in the sequence.
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Compute $S_4$ for $a_1=3$ and $r=-2$.
Compute $S_4$ for $a_1=3$ and $r=-2$.
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$S_4=3\frac{1-(-2)^4}{1-(-2)}=-15$. Use formula with negative ratio $r=-2$ and $n=4$.
$S_4=3\frac{1-(-2)^4}{1-(-2)}=-15$. Use formula with negative ratio $r=-2$ and $n=4$.
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Identify the common ratio $r$ of the geometric sequence $7,21,63,\cdots$.
Identify the common ratio $r$ of the geometric sequence $7,21,63,\cdots$.
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$r=3$. Divide consecutive terms: $\frac{21}{7}=3$ and $\frac{63}{21}=3$.
$r=3$. Divide consecutive terms: $\frac{21}{7}=3$ and $\frac{63}{21}=3$.
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Find $a_5$ for a geometric sequence with $a_1=2$ and $r=3$.
Find $a_5$ for a geometric sequence with $a_1=2$ and $r=3$.
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$a_5=2\cdot 3^4=162$. Use $a_n=a_1r^{n-1}$ with $n=5$, $a_1=2$, $r=3$.
$a_5=2\cdot 3^4=162$. Use $a_n=a_1r^{n-1}$ with $n=5$, $a_1=2$, $r=3$.
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Find the common ratio $r$ for the sequence $81,27,9,\cdots$.
Find the common ratio $r$ for the sequence $81,27,9,\cdots$.
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$r=\frac{1}{3}$. Divide consecutive terms: $\frac{27}{81}=\frac{1}{3}$.
$r=\frac{1}{3}$. Divide consecutive terms: $\frac{27}{81}=\frac{1}{3}$.
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Compute $S_6$ for $a_1=4$ and $r=\frac{1}{2}$.
Compute $S_6$ for $a_1=4$ and $r=\frac{1}{2}$.
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$S_6=4\frac{1-(\frac{1}{2})^6}{1-\frac{1}{2}}=\frac{63}{8}$. Apply formula with $a_1=4$, $r=\frac{1}{2}$, and $n=6$.
$S_6=4\frac{1-(\frac{1}{2})^6}{1-\frac{1}{2}}=\frac{63}{8}$. Apply formula with $a_1=4$, $r=\frac{1}{2}$, and $n=6$.
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Find $n$ if $a_1=3$, $r=2$, and $a_n=96$.
Find $n$ if $a_1=3$, $r=2$, and $a_n=96$.
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$n=6$. Use $a_n=a_1r^{n-1}$ to solve: $96=3\cdot 2^{n-1}$.
$n=6$. Use $a_n=a_1r^{n-1}$ to solve: $96=3\cdot 2^{n-1}$.
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Find $r$ if $a_1=2$, $a_4=54$, and the sequence is geometric.
Find $r$ if $a_1=2$, $a_4=54$, and the sequence is geometric.
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$r=3$. Use $a_4=a_1r^3$ to solve: $54=2r^3$, so $r^3=27$.
$r=3$. Use $a_4=a_1r^3$ to solve: $54=2r^3$, so $r^3=27$.
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Find $a_1$ if $a_3=20$ and $r=2$ for a geometric sequence.
Find $a_1$ if $a_3=20$ and $r=2$ for a geometric sequence.
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$a_1=\frac{20}{2^2}=5$. Use $a_3=a_1r^2$ to solve: $20=a_1\cdot 2^2$.
$a_1=\frac{20}{2^2}=5$. Use $a_3=a_1r^2$ to solve: $20=a_1\cdot 2^2$.
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Compute $S_3$ for $a_1=6$ and $r=-\frac{1}{2}$.
Compute $S_3$ for $a_1=6$ and $r=-\frac{1}{2}$.
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$S_3=6\frac{1-(-\frac{1}{2})^3}{1+\frac{1}{2}}=\frac{9}{2}$. Apply formula with $a_1=6$, $r=-\frac{1}{2}$, and $n=3$.
$S_3=6\frac{1-(-\frac{1}{2})^3}{1+\frac{1}{2}}=\frac{9}{2}$. Apply formula with $a_1=6$, $r=-\frac{1}{2}$, and $n=3$.
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Compute $S_2$ for $a_1=9$ and $r=\frac{2}{3}$.
Compute $S_2$ for $a_1=9$ and $r=\frac{2}{3}$.
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$S_2=9\frac{1-(\frac{2}{3})^2}{1-\frac{2}{3}}=15$. Apply formula with $a_1=9$, $r=\frac{2}{3}$, and $n=2$.
$S_2=9\frac{1-(\frac{2}{3})^2}{1-\frac{2}{3}}=15$. Apply formula with $a_1=9$, $r=\frac{2}{3}$, and $n=2$.
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Identify the error: using $S_n=a_1\frac{1-r^n}{1-r}$ when $r=1$. What is the correct sum?
Identify the error: using $S_n=a_1\frac{1-r^n}{1-r}$ when $r=1$. What is the correct sum?
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Use $S_n=na_1$ when $r=1$. When $r=1$, the geometric sum formula has zero denominator.
Use $S_n=na_1$ when $r=1$. When $r=1$, the geometric sum formula has zero denominator.
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What is the common ratio $r$ in a geometric sequence in terms of consecutive terms?
What is the common ratio $r$ in a geometric sequence in terms of consecutive terms?
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$r=\frac{a_{n+1}}{a_n}$. The ratio of any term to the previous term gives the common ratio.
$r=\frac{a_{n+1}}{a_n}$. The ratio of any term to the previous term gives the common ratio.
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What is the definition of a geometric sequence using first term $a_1$ and ratio $r$?
What is the definition of a geometric sequence using first term $a_1$ and ratio $r$?
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$a_n=a_1r^{n-1}$. Each term is the first term multiplied by $r$ raised to the term position minus 1.
$a_n=a_1r^{n-1}$. Each term is the first term multiplied by $r$ raised to the term position minus 1.
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What is the $n$th term $a_n$ in terms of $a_1$ and $r$ for a geometric sequence?
What is the $n$th term $a_n$ in terms of $a_1$ and $r$ for a geometric sequence?
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$a_n=a_1r^{n-1}$. General term formula where position determines the power of $r$.
$a_n=a_1r^{n-1}$. General term formula where position determines the power of $r$.
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If a loan has term $t$ years with monthly payments, what is the number of payments $n$?
If a loan has term $t$ years with monthly payments, what is the number of payments $n$?
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$n=12t$. Multiply years by 12 payments per year.
$n=12t$. Multiply years by 12 payments per year.
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What condition must hold to divide by $(1-r)$ when deriving the sum formula?
What condition must hold to divide by $(1-r)$ when deriving the sum formula?
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$r\ne 1$. Division by zero occurs if $r=1$, making the formula undefined.
$r\ne 1$. Division by zero occurs if $r=1$, making the formula undefined.
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