Geometric Sequences and Logarithms

[Astronomer107]

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!

 

[dduardo]

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]

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

 

[Astronomer107]

THANK YOU!