Geometric progression, is this correct? ratio/first/next term

In summary, the conversation discusses finding a formula for the expression 2^n - 2^{n-1} + 2^{n-2} - 2^{n-3} + ... + (-1)^{n-1} * 2 + (-1)^n when n is an integer greater than or equal to 1. The participants discuss the confusion about which is the first and last term, and eventually arrive at the formula \frac{2^{n}(1-(-\frac{1}{2})^{n+1}))}{\frac{3}{2}} for the sum of a geometric progression.
  • #1
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Hello everyone, I'm trying to solve the following:
If n is an integer and n > = 1, find a formula for the expression
[tex]2^n - 2^{n-1} + 2^{n-2} - 2^{n-3} + ... + (-1)^{n-1} * 2 + (-1)^n [/tex]

okay this confuses me, because I'm not sure which is the first term and which is the last term...
I figured the ratio was the following: -2
becuase if u take 2^n/[-2^(n-1)] = -2
I took the first term and divided it by the 2nd term to find the ratio or is it vice versa? by taking the 2nd term and dividing it by the first, which would give the ratio of: -1/2?

once i find the ratio, i can find the term right after the last by multiplying the ratio by the last term in this case (-1)^n is the last term, and the first term is 2^n correct?

Thanks, once i get this figured out i cna find the formual for the sum.
 
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  • #2
You take the 2nd term and divide it by the first, so [tex] r = -\frac{1}{2} [/tex]
 
  • #3
Thanks courtrigrad
Okay so the answer I'm getting would be:
["Mythical Next term" - "First real term" ]/[ratio - 1]
thats the general formula to find the formula for the sum of a geometric progression and i come up with:

sum = [-.5*(-1)^n - 2^n]/(-3/2)

does that look correct to you? I'm not sure how i can clean that up though.
 
  • #4
[tex] \sum_{n=0}^{\infty} ar^{k} = \frac{a(1-r^{n+1})}{1-r} [/tex]

The last term is [tex] (-1)^{n-1} 2^{n} [/tex]

[tex] \frac{2^{n}(1-(-\frac{1}{2})^{n+1}))}{\frac{3}{2}} [/tex]
 
Last edited:
  • #5
thanks a ton! I now see that the (-1)^n was just changing the + to - and vice versa! :)
 

1. What is a geometric progression?

A geometric progression is a sequence of numbers where each term is found by multiplying the previous term by a constant number. This constant number is called the common ratio.

2. How do you find the common ratio in a geometric progression?

To find the common ratio in a geometric progression, divide any term by the previous term. The result will be the common ratio.

3. What is the first term in a geometric progression?

The first term in a geometric progression is the starting number in the sequence. It is usually denoted as a1.

4. How do you find the next term in a geometric progression?

The next term in a geometric progression can be found by multiplying the previous term by the common ratio. This process can be repeated to find any term in the sequence.

5. Can a negative number be a term in a geometric progression?

Yes, a negative number can be a term in a geometric progression. The common ratio can also be a negative number, which would result in alternating positive and negative terms in the sequence.

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