## kathert Group Title 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 one year ago one year ago

1. cwrw238 Group Title

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

2. theEric Group Title
3. kathert Group Title

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

4. theEric Group Title

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

5. cwrw238 Group Title

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

6. cwrw238 Group Title

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

7. kathert Group Title

So would the answer for the first one be 87384?

8. theEric Group Title

@kathert I agree with 87384 :)

9. cwrw238 Group Title

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

10. kathert Group Title

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

11. cwrw238 Group Title

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

12. theEric Group Title

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 Group Title

@theEric Thanks :)

14. cwrw238 Group Title

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 Group Title

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

16. theEric Group Title

Do it like this: r ^ (- 7)

17. theEric Group Title

No!

18. theEric Group Title

19. theEric Group Title

r ^ (1/7)

20. theEric Group Title

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

21. kathert Group Title

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

22. kathert Group Title

I got 39364 for number 2 is that right?

23. kathert Group Title

Okay okay how do i do the last two??

24. theEric Group Title

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

25. theEric Group Title

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 Group Title

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

27. theEric Group Title

Do you see how to get that?

28. kathert Group Title

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

29. theEric Group Title

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 Group Title

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

31. kathert Group Title

wait so n is -11,246??

32. theEric Group Title

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

33. kathert Group Title

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

34. theEric Group Title

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 Group Title

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

36. kathert Group Title

Ohhhhh okay! I get it now!

37. theEric Group Title

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

38. kathert Group Title

I got -11 1/5

39. theEric Group Title

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

40. kathert Group Title

okay

41. theEric Group Title
42. theEric Group Title

Maybe you just had some calculator error.

43. kathert Group Title

Oh I see what I did wrong

44. kathert Group Title

How would I go about starting the last one?

45. theEric Group Title

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 Group Title

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 Group Title

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 Group Title

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

49. kathert Group Title

@theEric

50. theEric Group Title

Sorry! Hi!

51. kathert Group Title

Its alright I disappeared for dinner so... haha

52. theEric Group Title

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 Group Title

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 Group Title

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

55. kathert Group Title

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

56. theEric Group Title

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

57. kathert Group Title

so 390625?

58. theEric Group Title

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

59. kathert Group Title

so that goes where i put the 78125

60. theEric Group Title

Yep!

61. theEric Group Title

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

62. theEric Group Title

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

63. kathert Group Title

I got 781,248 for my answer

64. theEric Group Title

I got the same! :) Congrats!

65. theEric Group Title