Deriving/Applying the Geometric Series Formula - Algebra 2

1. What is the first term $a_1$ in terms of $a_n$, $r$, and $n$ for a geometric sequence?

   $a_1=\frac{a_n}{r^{n-1}}$. Solve $a_n=a_1r^{n-1}$ for $a_1$ by dividing by $r^{n-1}$.

2. What is $S_n-rS_n$ for a geometric series with first term $a_1$ and ratio $r$?

   $S_n-rS_n=a_1-a_1r^n$. Most terms cancel, leaving only first and last terms.

3. Find $S_5$ for a geometric series with $a_1=\frac{1}{2}$ and $r=4$.

   $\frac{341}{2}$. Apply $S_5=\frac{1}{2}\frac{1-4^5}{1-4}$ with large ratio.

4. Find $S_5$ for $a_1=5$, $r= \frac{1}{2}$, and $n=5$.

   $\frac{155}{16}$. Apply $S_5=5 \frac{1-( \frac{1}{2})^5}{1- \frac{1}{2}}$ with fractional ratio.

5. State the finite geometric series sum formula for $r \neq 1$ using $a_1$, $r$, and $n$ (alternate form).

   $S_n=a_1\frac{r^n-1}{r-1}$. Equivalent form obtained by factoring out $-1$ from numerator and denominator.

6. Find the sum $S_4$ if the first term is $a_1=8$ and the last term is $a_4=1$ with $r=\frac{1}{2}$.

   $15$. Use $S_n=\frac{a_1-a_nr}{1-r}=\frac{8-1\cdot\frac{1}{2}}{1-\frac{1}{2}}$.

7. Find $PV$ if $P=500$, $i=0.05$, and $n=1$ using $PV=P\frac{1-(1+i)^{-n}}{i}$.

   $PV\approx476.19$. Apply present value formula with single payment scenario.

8. Find $S_4$ for the geometric series $6-3+\frac{3}{2}-\frac{3}{4}$.

   $\frac{15}{4}$. Apply formula with $a_1=6$, $r=-\frac{1}{2}$, $n=4$.

9. Find $S_3$ for the geometric series $\frac{1}{3}+\frac{1}{6}+\frac{1}{12}$.

   $\frac{7}{12}$. Apply formula with $a_1=\frac{1}{3}$, $r=\frac{1}{2}$, $n=3$.

10. Find $S_3$ for $a_1=\frac{2}{5}$, $r=\frac{5}{2}$, and $n=3$.

    $\frac{39}{10}$. Apply $S_3=\frac{2}{5}\frac{1-(\frac{5}{2})^3}{1-\frac{5}{2}}$ with ratio $>1$.

11. Identify $r$ if $S_2=12$ and terms are $a_1=4$ and $a_2=4r$ in a geometric series.

    $r=2$. From $S_2=a_1+a_2=4+4r=12$, solve $4r=8$ to get $r=2$.

12. State the finite geometric series sum formula for $r\ne^1$ using $a_1$, $r$, and $n$ (alternate form).

    $S_n=a_1\frac{r^n-1}{r-1}$. Equivalent form obtained by factoring out $-1$ from numerator and denominator.

13. Identify the error: Using $S_n=a_1\frac{1-r^n}{1-r}$ when $r=1$.

    Formula invalid; use $S_n=na_1$ when $r=1$. When $r=1$, the formula has division by zero; use $S_n=na_1$ instead.

14. Find $S_3$ for $a_1=-5$, $r=\frac{1}{2}$, and $n=3$.

    $-\frac{35}{4}$. Apply $S_3=(-5)\frac{1-(\frac{1}{2})^3}{1-\frac{1}{2}}$ with negative first term.

15. Find $S_4$ for $a_1=1$, $r=-1$, and $n=4$.

    $0$. With $r=-1$ and even $n$, alternating terms cancel completely.

16. What is the last term $a_n$ of a geometric sequence in terms of $a_1$, $r$, and $n$?

    $a_n=a_1r^{n-1}$. Formula for the $n$-th term of a geometric sequence.

17. Find the common ratio $r$ for the geometric sequence $4,12,36,\dots$.

    $3$. Each term is 3 times the previous: $\frac{12}{4}=3$.

18. Find $PV$ if $P=100$, $i=0.10$, and $n=2$ using $PV=P\frac{1-(1+i)^{-n}}{i}$.

    $PV\approx173.55$. Apply present value formula with higher interest rate.

19. State the geometric sum formula in terms of first term $a_1$ and last term $a_n$.

    $S_n=\frac{a_1-a_nr}{1-r}$. Substitute $a_n=a_1r^{n-1}$ into the standard formula.

20. What is $S_4$ for the geometric series $81+27+9+3$?

    $120$. Apply $S_4=81\frac{1-(\frac{1}{3})^4}{1-\frac{1}{3}}$ with decreasing terms.

21. What is the sum $1+r+r^2+\cdots+r^{n-1}$ for $r \neq 1$?

    $\frac{1-r^n}{1-r}$. Standard geometric series sum formula for powers of $r$.

22. Find $n$ if $a_1=2$, $r=3$, and $a_n=162$ for a geometric sequence.

    $n=5$. From $a_n=a_1r^{n-1}=162$ with $a_1=2$, $r=3$, solve for $n$.

23. Which symbol is commonly used for the sum of the first $n$ terms of a series?

    $S_n$. Standard notation for partial sums of series.

24. What is the common ratio $r$ in terms of consecutive terms $a_n$ and $a_{n-1}$?

    $r=\frac{a_n}{a_{n-1}}$. Ratio between consecutive terms is constant in geometric sequences.

25. What factor is pulled out when simplifying $S_n-rS_n$ to solve for $S_n$?

    $ (1-r)$S_n$ $. Factor $ (1-r) $ from the left side to isolate $ $S_n$ $.

26. What is the sum of $n$ terms if $r=1$ and each term equals $a_1$?

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

27. What is the present value of an annuity formula for payment $P$, rate $i$, and $n$ payments?

    $PV=P\frac{1-(1+i)^{-n}}{i}$. Sum of discounted payments using geometric series with $r=(1+i)^{-1}$.

28. What is the definition of a finite geometric series sum $S_n$ in summation notation?

    $S_n=\sum_{k=1}^{n}a_1r^{k-1}$. Sum of terms $a_1r^{k-1}$ from $k=1$ to $n$ in a geometric series.

29. Find $S_4$ for $a_1=-2$, $r=3$, and $n=4$.

    $-80$. Apply $S_4=(-2)\frac{1-3^4}{1-3}=(-2)\frac{-80}{-2}=-80$.

30. What is $rS_n$ if $S_n=a_1+a_1r+\cdots+a_1r^{n-1}$?

    $rS_n=a_1r+a_1r^2+\cdots+a_1r^n$. Multiplying $S_n$ by $r$ shifts each term up one power of $r$.

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

    $a_n=a_1r^{n-1}$. Each term multiplies the previous by ratio $r$, starting from $a_1$.

32. What is $S_6$ for the geometric series $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}$?

    $\frac{63}{32}$. Apply formula with $a_1=1$, $r=\frac{1}{2}$, $n=6$.

33. Find $S_4$ for $a_1=9$, $r=\frac{1}{3}$, and $n=4$.

    $\frac{40}{3}$. Apply $S_4=9\frac{1-(\frac{1}{3})^4}{1-\frac{1}{3}}$ with small ratio.

34. Find $S_4$ if $a_2=6$, $r=2$, and the series is geometric.

    $45$. Find $a_1$ from $a_2=a_1r=6$, then calculate $S_4$.

35. Find $S_3$ if the geometric sequence has $a_1=2$ and $a_3=18$.

    $26$. Find $r$ from $a_3=a_1r^2=18$, then calculate $S_3$.

36. Find the monthly interest rate $i$ if the annual APR is $6\%$ compounded monthly.

    $i=\frac{0.06}{12}=0.005$. Convert annual rate to monthly by dividing by 12 periods.

37. Identify the common ratio $r$ in the present value series $P+(P)(1+i)^{-1}+\cdots$.

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

38. Find $S_6$ for $a_1=3$, $r=2$, and $n=6$.

    $189$. Apply $S_6=3\frac{1-2^6}{1-2}=3\frac{-63}{-1}=189$.

39. Find $S_3$ for $a_1=10$, $r=\frac{1}{10}$, and $n=3$.

    $11.1$. Apply $S_3=10\frac{1-(0.1)^3}{1-0.1}$ with decimal ratio.

40. Find $S_5$ if $a_1=2$, $r=3$, and $n=5$.

    $242$. Direct application of $S_5=2\frac{1-3^5}{1-3}$ gives 242.

41. State the finite geometric series sum formula for $r\ne^1$ using $a_1$, $r$, and $n$.

    $S_n=a_1\frac{1-r^n}{1-r}$. Derived by multiplying by $r$, subtracting, and solving for $S_n$.

42. State the finite geometric series sum formula for $r e 1$ using $a_1$, $r$, and $n$.

    $S_n=a_1\frac{1-r^n}{1-r}$. Derived by multiplying by $r$, subtracting, and solving for $S_n$.

43. Find $S_5$ for $a_1=5$, $r=\frac{1}{2}$, and $n=5$.

    $\frac{155}{16}$. Apply $S_5=5\frac{1-(\frac{1}{2})^5}{1-\frac{1}{2}}$ with fractional ratio.

44. Find $a_1$ if $a_4=54$ and $r=3$ for a geometric sequence.

    $2$. From $a_4=a_1r^3=54$ with $r=3$, solve $a_1\cdot27=54$.

45. Find $S_3$ for $a_1=7$, $r=-2$, and $n=3$.

    $7$. Apply $S_3=7\frac{1-(-2)^3}{1-(-2)}$ with alternating signs.

46. What restriction on $r$ is required to use $S_n=a_1\frac{1-r^n}{1-r}$?

    $r \neq 1$. Prevents division by zero in the denominator $(1-r)$.

47. What is $S_5$ for the geometric series $1+2+4+8+16$?

    $31$. Powers of 2 series: $S_5=1\frac{1-2^5}{1-2}=31$.

48. Find $P$ if $PV=1000$, $i=0.01$, and $n=2$ using $P=PV\frac{i}{1-(1+i)^{-n}}$.

    $P\approx507.51$. Substitute values into payment calculation formula.

49. State the geometric sum formula in terms of first term $a_1$ and last term $a_n$ (alternate form).

    $S_n=\frac{a_nr-a_1}{r-1}$. Alternative form by factoring out $-1$ from numerator and denominator.

50. What is $S_5$ for the geometric series $2+6+18+54+162$?

    $242$. Apply $S_5=2\frac{1-3^5}{1-3}$ with $a_1=2$, $r=3$, $n=5$.

51. State the loan payment formula for principal $PV$, monthly rate $i$, and $n$ payments.

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

52. Find $S_3$ using $S_n=\frac{a_1-a_nr}{1-r}$ when $a_1=3$, $a_3=12$, and $r=2$.

    $21$. Substitute known values into alternate sum formula.

53. Find $P$ if $PV=1000$, $i=0.05$, and $n=1$ using $P=PV\frac{i}{1-(1+i)^{-n}}$.

    $P=1050$. With one payment, $P=PV(1+i)$ for simple interest.

54. Find $1+3+9+27+81$ by using $S_n=a_1\frac{1-r^n}{1-r}$.

    $121$. Apply formula with $a_1=1$, $r=3$, $n=5$ for powers of 3.

55. Identify the error: Using $r=1+i$ as the ratio in the present value geometric series for loan payments.

    Use $r=(1+i)^{-1}$ for present value discounting. Present value requires discount factor $r=(1+i)^{-1}$, not growth factor.

56. Find $PV$ if $P=100$, $i=0.01$, and $n=2$ using $PV=P\frac{1-(1+i)^{-n}}{i}$.

    $PV\approx197.04$. Substitute values into present value annuity formula.

57. Find $S_5$ for $a_1=1$, $r=-1$, and $n=5$.

    $1$. With $r=-1$ and odd $n$, alternating terms give net sum of first term.

Study these flashcards