## kryton1212 2 years ago a and b are +ve integers. a,-4,b form a geometric sequence, and -4,b,a form an arithmetric sequence. (a) Find the value of ab. (b) Find the values of a and b. (c) (i) Find the sum to infinity of the geometric sequence a,-4,b,... . (ii) Find the sum to infinity of all the terms that are +ve in the geometric sequence a,-4,b.

1. kryton1212

@hartnn

2. hartnn

if x,y,z form an geometric sequence, then $$y^2 =xz$$ so, what about a,-4 and b ?

3. kryton1212

ab=(-4)^2=16 why?

4. kryton1212

it is also fixed?

5. hartnn

i can get you a simple explanation here... each next term of geometric sequence is MULTIPLIED by the common ratio 'r' so, in general, terms are a,ar,ar^2,ar^3..... when we have just 3 terms, we have a, ar, ar^2 for this (1st term * 3rd term) = (2nd term)^2 (you can check)

6. kryton1212

okay, i understand

7. hartnn

or in other words, they form a proportion, $$\dfrac{1st \: term}{2nd \: term}=\dfrac{2nd \: term}{3rd\: term}$$

8. hartnn

16 is correct :) now if the terms are in arithmetic sequence, then 2* 2nd term = 1st term + 3rd term use this...

9. kryton1212

wow, i like the formulae

10. hartnn

or if you don't want to remember the formula, you should know that difference between the terms is constant in arithmetic sequence so, difference between b and -4 = difference between a and b b - (-4) = a-b you'll get same equation :)

11. kryton1212

yup, got it

12. hartnn

tell me what u get for a and b when you get it....2 equations, 2 unknowns...

13. kryton1212

-4+d=b b+d=a ar=-4 ar^2=b are the equations correct?

14. hartnn

ar = -4 ? how ?

15. kryton1212

i don't know, i just guess.....it's written couple of days ago. i forgot how...

16. hartnn

forget about the 'd' and 'r' now we have 2 equations in a and b ab = 16 and from b-(-4) = a-b, we have 2b =a-4 you got these ? how..

17. kryton1212

uh, you said this moment ago. got it. a=8 , b=2

18. hartnn

those values are correct. so b) is done right ? now c) part can you find the common ratio in 8,-4,2,....?

19. kryton1212

*-2

20. hartnn

b=2 was correct....

21. kryton1212

common ratio is -2

22. hartnn

ohh! yes, r= -2 :) first term = a1 = 8 SUm to infinite is just $$\large \dfrac{a_1}{r-1}$$

23. kryton1212

what? -8/3?

24. hartnn

ohh....the r is not -2 :P

25. kryton1212

-1/2 ?

26. hartnn

yes. r = 2nd term / 1st term = 3rd trm/ 2nd term =.... = -4/8 = -1/2

27. kryton1212

-16/3 ...

28. hartnn

since r is LESS than 1, we use $$\huge S_\infty = \dfrac{a_1}{1-r}$$

29. kryton1212

lol

30. hartnn

yes, there are 2 formula, depending on r less than 1 or greater than 1

31. kryton1212

i will remember. I just forgot at that moment..

32. hartnn

try the last aprt ?

33. kryton1212

i hate doing +ve / -ve ....

34. hartnn

remove all the negatives... 8,-4,2,-1,1/2 .... to 8,2,1/2 .... now whats the common ratio r =... ?

35. kryton1212

oh, i know what's the world happening! r=1/4

36. hartnn

yessss!

37. hartnn

and since 1/4 is less than 1 you use a/ (1-r)

38. kryton1212

32/3

39. kryton1212

million thanks

40. hartnn

$$\huge \color{red}\checkmark$$

41. hartnn

welcome ^_^