kathert 2 years ago Please help... Only 2 days left for me on a summer school class and I am lost... 1. What is the sum of the geometric sequence 8, -16, 32 . if there are 15 terms? (1 point) 2. What is the sum of the geometric sequence 4, 12, 36 . if there are 9 terms? (1 point) 3. What is the sum of a 6-term geometric sequence if the first term is 11, the last term is -11,264 and the common ratio is -4? (1 point) 4. What is the sum of an 8-term geometric sequence if the first term is 10 and the last term is 781,250? (1 point) Show all work as well

1. cwrw238

the formula for sum of n terms is Sn = a1 * (1 - r^n) ------ 1 - r

2. theEric
3. kathert

But what do I plug in there to get the answers?

4. theEric

The link I posted will walk you through what geometric sequences are, and that formula to get the answers!

5. cwrw238

r = common ratio = second term / first term a1 = first term and n = number of terms

6. cwrw238

so for question 1 r = -16/8 = -2 a1 = 8 and n = 15

7. kathert

So would the answer for the first one be 87384?

8. theEric

@kathert I agree with 87384 :)

9. cwrw238

8 * ( 1 - (-2)^15) ------------- = 87384 1 - (-2)

10. kathert

Oh yay! I got it right! Sorry I suck at math...

11. cwrw238

A GS can be written as a, ar , ar^2 , ar^3 .......

12. theEric

I've $$always$$ been slow at math. But it took practice, and getting help, and I can do more math things now! Best of luck in your class! :) And it looks like you're in good hands with cwrw238 .

13. kathert

@theEric Thanks :)

14. cwrw238

for the last probem you can find the common ratio r by dividing the 8th term by the first then you take the 7th root 8th term = ar^7 8th term / first term = ar^7 / a = r^7 781,250 / 10 = 78125 now use your calculator to find the 7th root of 78125 then use the sum formula gotta go now

15. kathert

Whoa how do I do 7 root in my calculator??

16. theEric

Do it like this: r ^ (- 7)

17. theEric

No!

18. theEric

19. theEric

r ^ (1/7)

20. theEric

That's what you want, $\Huge r^{\frac{1}{7}}$

21. kathert

Haha okay gimme a minute to figure this out...

22. kathert

I got 39364 for number 2 is that right?

23. kathert

Okay okay how do i do the last two??

24. theEric

That's what I got for #2 as well.

25. theEric

3. What is the sum of a 6-term geometric sequence if the first term is 11, the last term is -11,264 and the common ratio is -4? (1 point) Well, the first term is your $$a$$.$a=11$The last term is your $$a\ r^n$$.$a\ r^n=-11,264$The common ratio is your $$r$$.$r=-4$ You need $$\Large a\frac{1-r^n}{1-r}$$.

26. theEric

So, you have $$a$$ and $$r$$, and you need $$n$$ or $$r^n$$.

27. theEric

Do you see how to get that?

28. kathert

uhhh.... no not really...so it would be -11 (1-(-4^n)/(1--4)?

29. theEric

Yep! Hey, you know $$a\ r^n=-11,264$$, so you can solve for $$r^n$$! That's how you'll finish that problem.

30. theEric

General guideline: if you want something, solve for it.

31. kathert

wait so n is -11,246??

32. theEric

No, $$a\ r^n=-11,264$$. So you divide both sides by $$a=11$$. That's algebra!

33. kathert

im sorry but im so lost.... where does the -11246 come in? do i set it equal to the equation?

34. theEric

It is necessary to find the $$r^n$$. Let me show you. Are you familiar with algebra? $a\ r^n = -11,264$and$a=11$By substituting $$11$$ for $$a$$, which is okay because it's the same value either way, you'll get:$11\ r^n=-11,264$Now, you want to get $$r^n$$ alone. So what you do is, you divide both sides by $$11$$. 1. If the two sides are equal, and you do the same thing to both sides, both sides will still be equal! 2. Why divide by $$11$$? Well $$r^n$$ is being multiplied by $$11$$, and so you want to negate that. You want to make it be $$\Large \frac{\cancel{11}\ r^n}{\cancel{11}}$$ So, we left our equation off at$11\ r^n=-11,264$We divide by $$11$$ to get$\frac{11\ r^n}{11}=\frac{-11,264}{11}$$\frac{\cancel{11}\ r^n}{\cancel{11}}=\frac{-11,264}{11}$$r^n=\frac{-11,264}{11}$

35. theEric

Since you now have $$r^n$$, $$a$$, and $$r$$, you can use that formula that you used for #1 and #2.

36. kathert

Ohhhhh okay! I get it now!

37. theEric

SWEET! :) So, we'll both calculate #3 and see what we get....

38. kathert

I got -11 1/5

39. theEric

I got $$2255$$... Let me use Wolfram Alpha to double check. Then I can show you a link to the math.

40. kathert

okay

41. theEric
42. theEric

Maybe you just had some calculator error.

43. kathert

Oh I see what I did wrong

44. kathert

How would I go about starting the last one?

45. theEric

Well, I'm sure you know the formula you need to use, by now!$\text{sum}=a\frac{(1-r^n)}{(1-r)}$ 4. What is the sum of an 8-term geometric sequence if the first term is 10 and the last term is 781,250? (1 point) You need $$a$$, $$r$$, and $$r^n$$, or $$n$$. What do you know from the problem, about the geometric sequence?

46. theEric

Refresher: $$a$$ is the first term, or the common multiplier. $$r$$ is the common ratio. $$n$$ is the number of terms in the sequence.

47. theEric

8-term $$\rightarrow n=8$$ first term is 10 $$\rightarrow a=10$$ last term is 781,250 $$\rightarrow a\ r^{n-1} =781,250$$

48. kathert

so would it be 10 (1-(781250/10))/1-r?

49. kathert

@theEric

50. theEric

Sorry! Hi!

51. kathert

Its alright I disappeared for dinner so... haha

52. theEric

Nope, sorry! Small mistake! $a\ r^{n-1} =781,250$$$\qquad\Downarrow\qquad$$Substitute $$8$$ for $$n$$ $a\ r^{8-1} =a\ r^{7}=781,250$$$\qquad\Downarrow\qquad$$Divide both sides by $$a$$, and then substitute $$10$$ in for $$a$$ $r^7=\frac{781,250}{a}=\frac{781,250}{10}=78,125$$$\qquad\Downarrow\qquad$$Get the seventh root of both sides$\sqrt[7]{r^7}=r=\sqrt[7]{78,125}=5$

53. theEric

Well, you knew you didn't know $$r$$, so I guess your only mistake was substituting $$r^n$$ with $$r^{n-1}=781,250$$, but you definitely had the right idea otherwise! I just found $$r$$ for you, then... Any questions on that part? Now you have $$a$$, $$n$$, and $$r$$.

54. theEric

Which you can rearrange to spell $$r\ a\ n$$: fun fact..

55. kathert

so it would be 8 *(1-78125)/(1-5) just to be sure nice fun fact btw haha

56. theEric

Haha, thanks! And check your "$$r^n$$", or $$5^8$$.

57. kathert

so 390625?

58. theEric

You got $$78125$$ from $$r^{n-1}$$, so I see where that came from :) And, yep! $$390625$$.

59. kathert

so that goes where i put the 78125

60. theEric

Yep!

61. theEric

It is $$r^n$$, after all.

62. theEric

and your formula is$\text{sum}=a\frac{(1-r^n)}{(1-r)}$

63. kathert

I got 781,248 for my answer

64. theEric

I got the same! :) Congrats!

65. theEric

66. kathert

Nope I'm good! Thank you for your help!

67. theEric

You're welcome! Take care!