kryton1212 2 years ago The general term of an infinite geometric sequence is given by: T(n)=6[(x+1)/8)^(n) (a) Find the range of the values of x for which the sum to infinity exists. (b) Find the sum to infinity of the sequence in terms of x. (c) If the sum to infinity of the sequence is 18, find the value of x.

1. kryton1212

$T(n)=6(\frac{ x+1 }{ 8 })^{n}$

2. kryton1212

@hartnn @Callisto

3. hartnn

the sum to infinity will converge(exist) if $$|r|<1$$ can you find 'r' from your general term ?

4. hartnn

general equation, $$T(n)=a_1r^{n-1}$$

5. kryton1212

umm..

6. hartnn

find a1 and 'r' both from your general term, you will need both

7. kryton1212

'r' is the ratio?

8. primeralph

|dw:1372921816382:dw|

9. kryton1212

??

10. hartnn

compare, try to make the power as (n-1) $$\large 6 (\dfrac{x+1}{8})^n = 6 \dfrac{x+1}{8}(\dfrac{x+1}{8})^{n-1}$$ now compare this with a1 r^{n-1} find a1 and r

11. hartnn

|dw:1372923030703:dw|

12. hartnn

then for the range of values of x, just do |r| <1

13. kryton1212

ahhh...

14. hartnn

@Jhannybean what! ?

15. primeralph

@Jhannybean Why?

16. hartnn

ohh..because @primeralph wrote "and" there! its actually "or"

17. primeralph

|dw:1372923436567:dw|

18. primeralph

@Jhannybean The x is only a part of r.

19. Jhannybean

Oh nevermind.... I was looking at it the wrong way.

20. kryton1212

i am still confusing..

21. Jhannybean

No i wasn't referring to that.

22. hartnn

@kryton1212 where are you confused ?

23. hartnn

you got this much ? $$\large |\dfrac{x+1}{8}|<1$$ ?

24. kryton1212

everywhere... i still cannot get what you mean..

25. kryton1212

what does | this line mean

26. hartnn

the absolute value sign

27. kryton1212

oh i see..

28. hartnn

$$if \quad |a|<b \\ a<b \quad or \quad a>-b$$

29. kryton1212

okay..

30. hartnn

so, what about |(x+1)/8| <1 ?

31. kryton1212

what does it mean

32. Jhannybean

OH its an OR. OK that makes more sense now, Got it.

33. hartnn

$$if \quad |a|<b \\ a<b \quad or \quad a>-b$$ $$\large if \quad |\dfrac{x+1}{8}|<1 \\ \large \dfrac{x+1}{8}<1 \quad or \quad \dfrac{x+1}{8}>-1$$

34. hartnn

now can you get the range ?

35. Jhannybean

So just solve for two x-values,and see which x-value fits the description of r<1.

36. kryton1212

-9<x<7

37. kryton1212

is it..

38. hartnn

yes, correct, thats your 1st part :)

39. kryton1212

thanks:) i got it ;D and part b.

40. hartnn

sum to infinity = $$\large \dfrac{a_1}{1-r}$$

41. hartnn

we have both, a1 and r

42. kryton1212

a1=?

43. hartnn

|dw:1372924494692:dw|

44. hartnn

$$\large a_1 =\dfrac{6(x+1)}{8}$$

45. hartnn

could you find the sum to infinity ?

46. kryton1212

|dw:1372925264670:dw|

47. hartnn

The general term in an geometric series is $$T(n)=a_1r^{n-1}$$ where, a1 is the 1st term of the series (which we get when we put n=1 in the general term) and 'r' is the common ratio, (ratio of next term to current term)

48. kryton1212

a1 and r...okay

49. hartnn

if you want more details about geometric sequence, you can go through this when u have time, http://openstudy.com/study#/updates/503bb2a0e4b007f9003103b0

50. hartnn

any more doubts ? what u got as sum to infinity ?

51. kryton1212

|dw:1372925634484:dw|

52. Jhannybean

Yay!! now simplify :)

53. hartnn

|dw:1372925676770:dw|

54. kryton1212

oh yeah

55. kryton1212

6(x+1) / 7-x

56. hartnn

correct! now for the last part, the sum is given as 18 so, getting x from 6 (x+1) / (7-x) = 18 is just simple algebra :)

57. Jhannybean

$\large 6 (\dfrac{x+1}{8})^n = 6 \dfrac{x+1}{8}(\dfrac{x+1}{8})^{n-1}$$\large a_{1}=\cfrac{6(x+1)}{8} \ \text{and} \ r = \left(\frac{x+1}{8}\right)$$\large s_n= \cfrac{ \cfrac{6(x+1)}{8}}{\cfrac{8-(x+1)}{8}}$

58. hartnn

to give you a head start, multiply both sides by 7-x

59. Jhannybean

$\large s_{n} = \cfrac{3(x+1)}{4}\cdot \cfrac{8}{-x+7} = \frac{6(x+1)}{-x+7}$

60. kryton1212

x=5

61. kryton1212

thanks @hartnn and @Jhannybean :)

62. Jhannybean

So yeah...you equal that equation to 18 and solve for x :) You stated that x=5, let's check that! $\large 18 = \cfrac{6(5+1)}{-5+7}$$\large 18=18 \ \checkmark$ good job!

63. hartnn

6x+6 = 126-18x 24x = 120 x=5 is correct :)