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Geometric Sequences and Logarithms

I'm having trouble with these type of probles (where a negative log comes up):

(All of this is solving without sigma notation)

Find the number of terms in these geometric sequences and the sum of the numbers.

11, -22, 44,....,704

I know that a1 = 11, r = -2, and an = 704, so I did:

704 = 11(-2)^n-1 so,

64 = (-2)^n-1

log 64 = (log -2) n-1
n should = 7, but when I found the log of 64 divided by the log of -2, I got .2785219413 - 1.2...

Why is there a minus sign and what am I doing wrong? Please help. Thanks!
 
Last edited:

dduardo

Staff Emeritus
1,894
3
Ignore the fact that the ratio is negative, since it will cause problems with the logarithm. (You can't take the log of a negative number)

the equation should start out as

704 = 11*2^x

so basically you get

11 * 2 * 2 * 2 * ..... = 704

you can simpliffy the equation by dividing both sides of the equation by 11

2^x = 64

x = ln(64)/ln(2) = 6

Then you have to add 1 to account for the first term

therefore the number of terms is 7
 

Hurkyl

Staff Emeritus
Science Advisor
Gold Member
14,845
17
You don't just ignore the negative sign; you take the absolute value of both sides.

(and because it's a nonreversible operation, you mave to check your answer to make sure it works)



And, incidentally, you can take the logarithm of a negative number, but it does require you to take a step up to complex analysis... the logarithm of a negative number is a (nonunique) complex number, and it takes a lot of care to make sure you are doing everything right.

When you computed log 64 / log -2, the result is thus a complex number and your calculator reported the principal value of this expression. If you had scrolled your display to the right, you would have seen 'i' in the answer
 
THANK YOU!!!!!!!!
 

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