## anonymous one year ago Medal and FAN John has taken out a loan for college. He started paying off the loan with a first payment of $100. Each month he pays, he wants to pay back 1.1 times as the amount he paid the month before. . Explain why this series is convergent or divergent • This Question is Closed 1. anonymous just tell me is this convergent or divergent 2. SolomonZelman well, just paying 1.1 times the previous payment means that sequence diverges (nothing to even talk about a series). BUT, a loan (I would think so) is some finite amount though, so this situtation with a loan is a bad example..... 3. SolomonZelman he won't go into paying more than the loan...... (and this all I am assuming there is no interest, since it is not mentioned here) 4. SolomonZelman So, lets write this series, and lets diregard the fact that he is to pay some finite loan amount. OK? 5. anonymous is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. 6. anonymous is divergent 7. SolomonZelman $$\large\color{black}{ \displaystyle a_1=100 }$$ $$\large\color{black}{ \displaystyle r=1.1 }$$ and this is a perfect geometric series. For geometric series to converge, normally, the folloing mst be true: |r|<1 yes correct (disregarding the fact that it's a loan). 8. SolomonZelman The fact that it is a loan is forcing a FINITE series. 9. SolomonZelman If you were to propose a different scenario, such as: 10. SolomonZelman John is giving a charity to poor children who stand by the Church. THe first month he gave 100$ and then, every month, he wants to give 1.1 times as much as he gave the previous month. is this series convergent or divergent?

11. SolomonZelman

this would be more like an INFINITE series scenraio....

12. anonymous

ok thanks can u help with another question

13. SolomonZelman

ok, but what course is this?

14. SolomonZelman

is this geometry or calc?

15. anonymous

algebra 2

16. SolomonZelman

17. SolomonZelman

((in calc there are more advanced "tools" to use))

18. anonymous

The graph of f(x) = 2^x + 1 is shown below. Explain how to find the average rate of change between x = 0 and x = 3

19. SolomonZelman

find f(0). find f(3). find the slope between (0,f(0)) and (3,f(3)).

20. SolomonZelman

making sense?

21. anonymous

22. anonymous

thats the graph

23. SolomonZelman

no need for a graph to perform this task, but, it is nice to have one.

24. SolomonZelman

do my steps to the solution make sense?

25. anonymous

no

26. SolomonZelman

Ok, lets do it togther one by one.

27. anonymous

28. SolomonZelman

yes

29. SolomonZelman

did you find f(0) and f(3) ?

30. SolomonZelman

f(x) = 2$$^x$$+1

31. SolomonZelman

f(0)=2$$^0$$+1=? f(3)=2$$^3$$+1=?

32. anonymous

f(3) =7

33. SolomonZelman

close

34. SolomonZelman

2³ = ?

35. anonymous

9

36. anonymous

oops

37. SolomonZelman

9 is f(3) or just the 2³ part?

38. anonymous

2^3

39. SolomonZelman

$$2^3=2\times 2 \times 2$$

40. SolomonZelman

=8

41. anonymous

f(0) = 1

42. anonymous

f(3) = 9

43. anonymous

i meant that

44. SolomonZelman

yes,

45. SolomonZelman

f(3)=9

46. SolomonZelman

and f(0)=?

47. anonymous

1

48. SolomonZelman

no

49. SolomonZelman

your function is $$f(x)=2^x+1$$

50. SolomonZelman

$$f(0)=2^0+1=?$$

51. anonymous

f(0) = 0 + 1

52. SolomonZelman

$${\rm C}^0=1$$ (for any real number C)

53. anonymous

f(0) = 1

54. SolomonZelman

$$f(0)=2^0+1=1+1=?$$

55. anonymous

2

56. SolomonZelman

yes, f(0)=2

57. anonymous

Explain how to find the average rate of change between x = 0 and x = 3.

58. SolomonZelman

now, you need to find the slope between (0,f(0)) and (3,f(3)). we know f(3)=9, f(0)=2. SO lets plug this in. ~ you have to find the slope between (0,2) and (3,9)

59. SolomonZelman

$$\LARGE \color{black}{ \displaystyle {\rm m}=\frac{\color{blue}{{\rm y}_1}-\color{red}{{\rm y}_2}}{\color{green}{{\rm x}_1}-\color{darkgoldenrod}{{\rm x}_2}} }$$ where $$\LARGE \color{black}{ \displaystyle {\rm m} }$$ is the slope $$\Large\color{black}{ \displaystyle (\color{green}{{\rm x}_1}~,~~\color{blue}{{\rm y}_1}) }$$ and $$\Large\color{black}{ \displaystyle (\color{darkgoldenrod}{{\rm x}_2}~,~~\color{red}{{\rm y}_2}) }$$ are your two points.

60. anonymous

9-2 =7 3-0 =3

61. SolomonZelman

62. anonymous

7/3

63. SolomonZelman

yup

64. anonymous

65. SolomonZelman

66. anonymous

so what is the average rate of change again

67. SolomonZelman

7/3 :)

68. SolomonZelman

slope and average rate of change (in this context) is the same thing.

69. anonymous

thanks so much

70. SolomonZelman

you are welcome !