question (1 Viewer)

[Xu]

Please help me. i know this is easy, but i cant do it!!

note: log_2 means a base 2

the sum of the first 10 terms of the sereis log_2(1/x) + log_2(1/x^2) + log_2(1/x^3) +... for x>0 is -440. find that vulue of x.

 

[Antisocial]

Hahaha, nice poll.
I will do this question as I am typing (need to practise series and sequences, anyway).

Test to see if AP or GP, ie. T2 - T1 = T3 - T2.

T2 - T1 = log_2 (1/x) - log_2 (1/x)

Hmm. Remember subtraction of two logs is division...

log_2 (1/x / 1/x) = log_2 (1/x)

Bring the bottom denominator up top, and simplify. Then we work on...

T3 - T2 = log_2 (1/x / 1/x) = log_2 (1/x)

log_2 (1/x) = T2 - T1 = T3 - T2

Therefore it's an AP. Some forward movement now.

It says sum of 10 terms is -440. So, sum of AP.. sn = n/2 [a + (n-1)d].

-440 = 10/2 [2log_x (1/x) + (10 - 1)log_2 (1/x)]

Expand and simplify...

10log_2 (1/x) + 45log_2 (1/x) = -440

Take out HCF...

log_2 (1/x) [10 + 45] = -440

Solve for x.

log_2 (1/x) = -440 / 55

log_2x = 8

x = 2^8
x = 256

That... took longer than expected.
Hope that's the right answer.

 

[Jumbo Cactuar]

this is a way better solution ...

-440 = log_2 (1/x) + ... + log_2 (1/x^10)
= log_2 ((1/x) * ... * (1/x^10))
= log_2 (1/x^55)
2^-440 = 1/x^55
2^440 = x^55

x = 2^(440/55) = 256

 

[Xu]

Originally posted by Jumbo Cactuar
this is a way better solution ...

-440 = log_2 (1/x) + ... + log_2 (1/x^10)
= log_2 ((1/x) * ... * (1/x^10))
= log_2 (1/x^55)
2^-440 = 1/x^55
2^440 = x^55

x = 2^(440/55) = 256

hi, sorry its me again. umm.. could soemone please explain to me who dumbo cactar got his/her (sorry) answer?

sorry.

 

[hereSIR]

RHS=log_2 (1/x) + ... + log_2 (1/x^10)
= log_2 ((1/x) * ... * (1/x^10)) (Log Rule-"the addition rule thing")

Now (1/x) * ... * (1/x^10) = (1/x^55) (Index Law - when multiply we add the powers, i.e 1+2+3......+10 = 55)
Hence the rest! (I think ).

 

[Jumbo Cactuar]

Originally posted by Xu
dumbo

fsss the outrage!!!

Well that is the last time I help a fellow ffviii enthusistast, or someone who coincidentally has the name of one of the characters

 

[ssssonicyouth]

Originally posted by Jumbo Cactuar
this is a way better solution ...

yes, but u missed so many lines working it was pretty pointless to use it as an attempt to explain something to xu