Summing Geometric Sequences am i doing this right?

[mr_coffee]

We learned in class how to find the sum of any geometric sequence with the following formula:

Let x = Sum Of Geometric Sequence;

x = [Take mythical next term - real term]/(ratio - 1);

The real term is the first term of the sequence and the mythical next term would be the next term for example after this sequence:

ex:
1+2+2^2+2^3 +...+2^n = ?

first term is : 1
mythical next term is: 2^(n+1)
ratio: 2
when I say ratio I mean what you multiply each term to get to the next and you can see its 2 here.

x = [2^(n+1)-1]/(2-1)

So that would be the formula to sum up the sequence...so that one was easy but this is the one that i don't understand:


2^n + 2^(n-1) x 3 + 2^(n-2) x 3^2 + 2^(n-3) x 3^3 + ... + 2^3 x 3^(n-3)

I don't know if I'm getting the ratio right or not...
I see each term is getting multiplied by 3, the first term is just 3^0 = 1, then 3^1, 3^2...

I see n is being decremented by 1 each time and its a power of 2, so would that be: 1/2^n

So the ratio i figured out is: 3/2^n

Now the mythical next term I got is: 2^4 x 3^(n-4)
or do i not mess with the 2^3? and leave it as 2^3 x 3^(n-4) ?

The 1st real term is: 2^n

So here is the formula i come out with:

Sum of geometric sequence = [ [2^4 x 3^(n-4)] - 2^n ]/[(3/2^n)-1]

I'm testing for n = 3, n = 4, and n = 5 to see if itss right...
for n = 3
2^3 + 2^2 x 3 + 2 x 3^2 = 38

now if i plug 3 into the formula:
[2^4 x 3^(-1) - 2^3 ] / (3/(2^3) - 1) = 64/15 which isn't 38...

Do you see what I'm doing wrong? I think I'm screwing up on the ratio and the mythical next term, the next term in the sequence, any help would be great. I have to do it the way the professor showed us with that general formula:

Let x = Sum Of Geometric Sequence;

x = [Take mythical next term - real term]/(ratio - 1);


Thanks!

 

[StatusX]

The ratio is just 3/2, not 3/2^n. Multiply the last term by this ratio to get the "mythical next term."

 

[mr_coffee]

Thanks for the replyoh, multiplying by the ratio would make sense

Okay now i have as the last term: 4/9 x 3^n as the next term
and ratio: 3/2

x = [ (4/9) x 3^n - 2^n ]/[(3/2) - 1]

if i plug in n = 3, i get x = 8

if i go back to the sequence and add up the first 3 terms, after letting n = 3, i get the following:

2^3 + 2^(2) x 3 + 2 x 3^2 = 38

Is this not how you check to see if the formula is correct or not? If i let n = 3 in the formula, it should sum the first 3 terms right?

 

[StatusX]

No, the power of 2 runs from n down to 3, so for n=3 there is only one term, 2^3=8.

 

[mr_coffee]

yep that works! thanks!

I think i get what your saying, so if I wanted to check for n = 4, i would just add up
2^4 + 2^(4-1) = 20

and 5 i would add up the first 3 terms with n = 5 right?

 

[StatusX]

Well that sum isn't right: it's missing the 3's, and 2^4+2^3=16+8=24. What you said about the number of terms is right, but why don't you just check the new formula you found to see if it works?

 

[mr_coffee]

alroight now i get it! hah sorry I'm coming down from a cafine overdose i think.