Deriving/Applying the Geometric Series Formula - Algebra 2
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Find $S_5$ if $a_1=2$, $r=3$, and $n=5$.
Find $S_5$ if $a_1=2$, $r=3$, and $n=5$.
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$242$. Direct application of $S_5=2\frac{1-3^5}{1-3}$ gives 242.
$242$. Direct application of $S_5=2\frac{1-3^5}{1-3}$ gives 242.
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What restriction on $r$ is required to use $S_n=a_1\frac{1-r^n}{1-r}$?
What restriction on $r$ is required to use $S_n=a_1\frac{1-r^n}{1-r}$?
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$r \neq 1$. Prevents division by zero in the denominator $(1-r)$.
$r \neq 1$. Prevents division by zero in the denominator $(1-r)$.
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Find $S_5$ for a geometric series with $a_1=\frac{1}{2}$ and $r=4$.
Find $S_5$ for a geometric series with $a_1=\frac{1}{2}$ and $r=4$.
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$\frac{341}{2}$. Apply $S_5=\frac{1}{2}\frac{1-4^5}{1-4}$ with large ratio.
$\frac{341}{2}$. Apply $S_5=\frac{1}{2}\frac{1-4^5}{1-4}$ with large ratio.
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Find the sum $S_4$ if the first term is $a_1=8$ and the last term is $a_4=1$ with $r=\frac{1}{2}$.
Find the sum $S_4$ if the first term is $a_1=8$ and the last term is $a_4=1$ with $r=\frac{1}{2}$.
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$15$. Use $S_n=\frac{a_1-a_nr}{1-r}=\frac{8-1\cdot\frac{1}{2}}{1-\frac{1}{2}}$.
$15$. Use $S_n=\frac{a_1-a_nr}{1-r}=\frac{8-1\cdot\frac{1}{2}}{1-\frac{1}{2}}$.
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Find $S_3$ for $a_1=\frac{2}{5}$, $r=\frac{5}{2}$, and $n=3$.
Find $S_3$ for $a_1=\frac{2}{5}$, $r=\frac{5}{2}$, and $n=3$.
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$\frac{39}{10}$. Apply $S_3=\frac{2}{5}\frac{1-(\frac{5}{2})^3}{1-\frac{5}{2}}$ with ratio $>1$.
$\frac{39}{10}$. Apply $S_3=\frac{2}{5}\frac{1-(\frac{5}{2})^3}{1-\frac{5}{2}}$ with ratio $>1$.
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Find $1+3+9+27+81$ by using $S_n=a_1\frac{1-r^n}{1-r}$.
Find $1+3+9+27+81$ by using $S_n=a_1\frac{1-r^n}{1-r}$.
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$121$. Apply formula with $a_1=1$, $r=3$, $n=5$ for powers of 3.
$121$. Apply formula with $a_1=1$, $r=3$, $n=5$ for powers of 3.
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Identify the error: Using $r=1+i$ as the ratio in the present value geometric series for loan payments.
Identify the error: Using $r=1+i$ as the ratio in the present value geometric series for loan payments.
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Use $r=(1+i)^{-1}$ for present value discounting. Present value requires discount factor $r=(1+i)^{-1}$, not growth factor.
Use $r=(1+i)^{-1}$ for present value discounting. Present value requires discount factor $r=(1+i)^{-1}$, not growth factor.
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Find $PV$ if $P=100$, $i=0.10$, and $n=2$ using $PV=P\frac{1-(1+i)^{-n}}{i}$.
Find $PV$ if $P=100$, $i=0.10$, and $n=2$ using $PV=P\frac{1-(1+i)^{-n}}{i}$.
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$PV\approx173.55$. Apply present value formula with higher interest rate.
$PV\approx173.55$. Apply present value formula with higher interest rate.
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Find $PV$ if $P=500$, $i=0.05$, and $n=1$ using $PV=P\frac{1-(1+i)^{-n}}{i}$.
Find $PV$ if $P=500$, $i=0.05$, and $n=1$ using $PV=P\frac{1-(1+i)^{-n}}{i}$.
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$PV\approx476.19$. Apply present value formula with single payment scenario.
$PV\approx476.19$. Apply present value formula with single payment scenario.
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Find $P$ if $PV=1000$, $i=0.05$, and $n=1$ using $P=PV\frac{i}{1-(1+i)^{-n}}$.
Find $P$ if $PV=1000$, $i=0.05$, and $n=1$ using $P=PV\frac{i}{1-(1+i)^{-n}}$.
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$P=1050$. With one payment, $P=PV(1+i)$ for simple interest.
$P=1050$. With one payment, $P=PV(1+i)$ for simple interest.
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Find $P$ if $PV=1000$, $i=0.01$, and $n=2$ using $P=PV\frac{i}{1-(1+i)^{-n}}$.
Find $P$ if $PV=1000$, $i=0.01$, and $n=2$ using $P=PV\frac{i}{1-(1+i)^{-n}}$.
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$P\approx507.51$. Substitute values into payment calculation formula.
$P\approx507.51$. Substitute values into payment calculation formula.
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Find $PV$ if $P=100$, $i=0.01$, and $n=2$ using $PV=P\frac{1-(1+i)^{-n}}{i}$.
Find $PV$ if $P=100$, $i=0.01$, and $n=2$ using $PV=P\frac{1-(1+i)^{-n}}{i}$.
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$PV\approx197.04$. Substitute values into present value annuity formula.
$PV\approx197.04$. Substitute values into present value annuity formula.
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Identify the common ratio $r$ in the present value series $P+(P)(1+i)^{-1}+\cdots$.
Identify the common ratio $r$ in the present value series $P+(P)(1+i)^{-1}+\cdots$.
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$r=(1+i)^{-1}$. Each payment is discounted by factor $(1+i)^{-1}$ from previous.
$r=(1+i)^{-1}$. Each payment is discounted by factor $(1+i)^{-1}$ from previous.
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What is the present value of an annuity formula for payment $P$, rate $i$, and $n$ payments?
What is the present value of an annuity formula for payment $P$, rate $i$, and $n$ payments?
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$PV=P\frac{1-(1+i)^{-n}}{i}$. Sum of discounted payments using geometric series with $r=(1+i)^{-1}$.
$PV=P\frac{1-(1+i)^{-n}}{i}$. Sum of discounted payments using geometric series with $r=(1+i)^{-1}$.
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Find the monthly interest rate $i$ if the annual APR is $6%$ compounded monthly.
Find the monthly interest rate $i$ if the annual APR is $6%$ compounded monthly.
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$i=\frac{0.06}{12}=0.005$. Convert annual rate to monthly by dividing by 12 periods.
$i=\frac{0.06}{12}=0.005$. Convert annual rate to monthly by dividing by 12 periods.
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Find $S_5$ for a geometric series with $a_1=\frac{1}{2}$ and $r=4$.
Find $S_5$ for a geometric series with $a_1=\frac{1}{2}$ and $r=4$.
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$\frac{341}{2}$. Apply $S_5=\frac{1}{2}\frac{1-4^5}{1-4}$ with large ratio.
$\frac{341}{2}$. Apply $S_5=\frac{1}{2}\frac{1-4^5}{1-4}$ with large ratio.
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Find $S_4$ if $a_2=6$, $r=2$, and the series is geometric.
Find $S_4$ if $a_2=6$, $r=2$, and the series is geometric.
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$45$. Find $a_1$ from $a_2=a_1r=6$, then calculate $S_4$.
$45$. Find $a_1$ from $a_2=a_1r=6$, then calculate $S_4$.
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Find $S_3$ if the geometric sequence has $a_1=2$ and $a_3=18$.
Find $S_3$ if the geometric sequence has $a_1=2$ and $a_3=18$.
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$26$. Find $r$ from $a_3=a_1r^2=18$, then calculate $S_3$.
$26$. Find $r$ from $a_3=a_1r^2=18$, then calculate $S_3$.
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Identify $r$ if $S_2=12$ and terms are $a_1=4$ and $a_2=4r$ in a geometric series.
Identify $r$ if $S_2=12$ and terms are $a_1=4$ and $a_2=4r$ in a geometric series.
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$r=2$. From $S_2=a_1+a_2=4+4r=12$, solve $4r=8$ to get $r=2$.
$r=2$. From $S_2=a_1+a_2=4+4r=12$, solve $4r=8$ to get $r=2$.
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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 multiplies the previous by ratio $r$, starting from $a_1$.
$a_n=a_1r^{n-1}$. Each term multiplies the previous by ratio $r$, starting from $a_1$.
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What is the definition of a finite geometric series sum $S_n$ in summation notation?
What is the definition of a finite geometric series sum $S_n$ in summation notation?
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$S_n=\sum_{k=1}^{n}a_1r^{k-1}$. Sum of terms $a_1r^{k-1}$ from $k=1$ to $n$ in a geometric series.
$S_n=\sum_{k=1}^{n}a_1r^{k-1}$. Sum of terms $a_1r^{k-1}$ from $k=1$ to $n$ in a geometric series.
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State the finite geometric series sum formula for $r\ne^1$ using $a_1$, $r$, and $n$.
State the finite geometric series sum formula for $r\ne^1$ using $a_1$, $r$, and $n$.
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$S_n=a_1\frac{1-r^n}{1-r}$. Derived by multiplying by $r$, subtracting, and solving for $S_n$.
$S_n=a_1\frac{1-r^n}{1-r}$. Derived by multiplying by $r$, subtracting, and solving for $S_n$.
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State the finite geometric series sum formula for $r\ne^1$ using $a_1$, $r$, and $n$ (alternate form).
State the finite geometric series sum formula for $r\ne^1$ using $a_1$, $r$, and $n$ (alternate form).
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$S_n=a_1\frac{r^n-1}{r-1}$. Equivalent form obtained by factoring out $-1$ from numerator and denominator.
$S_n=a_1\frac{r^n-1}{r-1}$. Equivalent form obtained by factoring out $-1$ from numerator and denominator.
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What is $rS_n$ if $S_n=a_1+a_1r+\cdots+a_1r^{n-1}$?
What is $rS_n$ if $S_n=a_1+a_1r+\cdots+a_1r^{n-1}$?
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$rS_n=a_1r+a_1r^2+\cdots+a_1r^n$. Multiplying $S_n$ by $r$ shifts each term up one power of $r$.
$rS_n=a_1r+a_1r^2+\cdots+a_1r^n$. Multiplying $S_n$ by $r$ shifts each term up one power of $r$.
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What is $S_n-rS_n$ for a geometric series with first term $a_1$ and ratio $r$?
What is $S_n-rS_n$ for a geometric series with first term $a_1$ and ratio $r$?
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$S_n-rS_n=a_1-a_1r^n$. Most terms cancel, leaving only first and last terms.
$S_n-rS_n=a_1-a_1r^n$. Most terms cancel, leaving only first and last terms.
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What is the sum of $n$ terms if $r=1$ and each term equals $a_1$?
What is the sum of $n$ terms if $r=1$ and each term equals $a_1$?
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$S_n=na_1$. When $r=1$, all terms equal $a_1$, so sum is $n$ times $a_1$.
$S_n=na_1$. When $r=1$, all terms equal $a_1$, so sum is $n$ times $a_1$.
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What is the last term $a_n$ of a geometric sequence in terms of $a_1$, $r$, and $n$?
What is the last term $a_n$ of a geometric sequence in terms of $a_1$, $r$, and $n$?
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$a_n=a_1r^{n-1}$. Formula for the $n$-th term of a geometric sequence.
$a_n=a_1r^{n-1}$. Formula for the $n$-th term of a geometric sequence.
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State the geometric sum formula in terms of first term $a_1$ and last term $a_n$.
State the geometric sum formula in terms of first term $a_1$ and last term $a_n$.
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$S_n=\frac{a_1-a_nr}{1-r}$. Substitute $a_n=a_1r^{n-1}$ into the standard formula.
$S_n=\frac{a_1-a_nr}{1-r}$. Substitute $a_n=a_1r^{n-1}$ into the standard formula.
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What factor is pulled out when simplifying $S_n-rS_n$ to solve for $S_n$?
What factor is pulled out when simplifying $S_n-rS_n$ to solve for $S_n$?
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$ (1-r)S_n $. Factor $ (1-r) $ from the left side to isolate $ S_n $.
$ (1-r)S_n $. Factor $ (1-r) $ from the left side to isolate $ S_n $.
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State the geometric sum formula in terms of first term $a_1$ and last term $a_n$ (alternate form).
State the geometric sum formula in terms of first term $a_1$ and last term $a_n$ (alternate form).
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$S_n=\frac{a_nr-a_1}{r-1}$. Alternative form by factoring out $-1$ from numerator and denominator.
$S_n=\frac{a_nr-a_1}{r-1}$. Alternative form by factoring out $-1$ from numerator and denominator.
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