We learned in class how to find the sum of any geometric sequence with the following formula:(adsbygoogle = window.adsbygoogle || []).push({});

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!

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# Summing Geometric Sequences am i doing this right?

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