Deriving/Applying the Geometric Series Formula - Algebra 1

1. Convert an annual interest rate $R$ compounded monthly to periodic rate $i$. What is $i$?

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

2. What is the sum of the first $n$ terms if $r=1$ and first term is $a_1$?

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

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

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

4. Compute $n$ for a $15$-year loan with monthly payments.

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

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

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

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

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

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

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

8. Find $a_1$ if $S_3=21$, $r=2$, and $n=3$.

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

9. Compute $S_4$ for $a_1=\frac{1}{3}$ and $r=3$.

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

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

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

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

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

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

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

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

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

14. Find $n$ if $a_1=3$, $r=2$, and $a_n=96$.

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

15. Find $a_5$ for a geometric sequence with $a_1=2$ and $r=3$.

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

16. What is the common ratio $r$ in a geometric sequence in terms of consecutive terms?

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

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

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

18. Find the sum $S_3$ for $a_1=1$ and $r=-1$.

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

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

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

20. What is the $n$th term $a_n$ in terms of $a_1$ and $r$ for a geometric sequence?

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

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

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

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

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

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

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

24. Find $a_1$ if $a_3=20$ and $r=2$ for a geometric sequence.

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

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

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

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

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

27. Compute $S_5$ for $a_1=1$ and $r=\frac{1}{2}$.

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

28. What is the key multiplication step used to derive the geometric sum (multiply $S_n$ by what)?

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

29. What is the last term $a_n$ in a geometric series expressed using $a_1$, $r$, and $n$?

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

30. If a loan has term $t$ years with monthly payments, what is the number of payments $n$?

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

31. Compute the sum $S_4$ for the series $5+15+45+135$.

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

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

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

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

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

34. Compute the monthly rate $i$ for an APR of $6\%$ compounded monthly.

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

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

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

36. In the discounted-payment series, what is the $n$th term $a_n$?

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

37. Compute $S_4$ for $a_1=3$ and $r=-2$.

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

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

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

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

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

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

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

41. What condition must hold to divide by $(1-r)$ when deriving the sum formula?

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

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

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

43. In the derivation, what expression results from subtracting: $S_n-rS_n$?

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

44. Find the common ratio $r$ for the sequence $81,27,9,\cdots$.

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

45. What is the loan payment formula for principal $PV$, periodic rate $i$, and $n$ payments?

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

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

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

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

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

48. What is the definition of a geometric sequence using first term $a_1$ and ratio $r$?

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

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

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

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

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

51. Compute $S_3$ for the geometric series $10+5+2.5$.

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

Study these flashcards