Geometric Sequences and Logarithms

In summary, the conversation is about solving for the number of terms and sum of numbers in a geometric sequence. The individual is having trouble with negative logs and is asking for help. The solution involves using absolute value and taking the logarithm of a negative number. The final answer is 7 terms.
  • #1
Astronomer107
31
0
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!
 
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  • #2
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
 
  • #3
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
 
  • #4
THANK YOU!
 

What is a geometric sequence?

A geometric sequence is a sequence of numbers where the ratio between consecutive terms is constant. This means that each term is multiplied by the same number to get the next term in the sequence.

How do you find the common ratio of a geometric sequence?

The common ratio of a geometric sequence can be found by dividing any term in the sequence by the previous term. For example, if the sequence is 2, 6, 18, 54, the common ratio is 6/2 = 3.

What is the formula for finding the nth term of a geometric sequence?

The formula for finding the nth term of a geometric sequence is:
an = a1 * rn-1
where a1 is the first term in the sequence and r is the common ratio.

What is a logarithm?

A logarithm is the inverse function of an exponential. It is used to solve equations where the variable is in the exponent.

How do you solve equations involving logarithms?

To solve an equation involving logarithms, you can use the laws of logarithms to simplify the equation and then solve for the variable. The laws of logarithms include the product rule, quotient rule, and power rule.

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