## kryton1212 Group Title 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. one year ago one year ago

1. kryton1212 Group Title

@hartnn

2. hartnn Group Title

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

3. kryton1212 Group Title

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

4. kryton1212 Group Title

it is also fixed?

5. hartnn Group Title

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 Group Title

okay, i understand

7. hartnn Group Title

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

8. hartnn Group Title

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

9. kryton1212 Group Title

wow, i like the formulae

10. hartnn Group Title

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 Group Title

yup, got it

12. hartnn Group Title

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

13. kryton1212 Group Title

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

14. hartnn Group Title

ar = -4 ? how ?

15. kryton1212 Group Title

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

16. hartnn Group Title

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 Group Title

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

18. hartnn Group Title

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

19. kryton1212 Group Title

*-2

20. hartnn Group Title

b=2 was correct....

21. kryton1212 Group Title

common ratio is -2

22. hartnn Group Title

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

23. kryton1212 Group Title

what? -8/3?

24. hartnn Group Title

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

25. kryton1212 Group Title

-1/2 ?

26. hartnn Group Title

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

27. kryton1212 Group Title

-16/3 ...

28. hartnn Group Title

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

29. kryton1212 Group Title

lol

30. hartnn Group Title

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

31. kryton1212 Group Title

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

32. hartnn Group Title

try the last aprt ?

33. kryton1212 Group Title

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

34. hartnn Group Title

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

35. kryton1212 Group Title

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

36. hartnn Group Title

yessss!

37. hartnn Group Title

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

38. kryton1212 Group Title

32/3

39. kryton1212 Group Title

million thanks

40. hartnn Group Title

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

41. hartnn Group Title

welcome ^_^