SOLUTION: The first and last term of a geometric series are 2 and 2048 respectively. The sum of the series is 273. Find the common ratio and the number of terms.

Algebra ->  Sequences-and-series -> SOLUTION: The first and last term of a geometric series are 2 and 2048 respectively. The sum of the series is 273. Find the common ratio and the number of terms.       Log On


   

Question 825369: The first and last term of a geometric series are 2 and 2048 respectively. The sum of the series is 273. Find the common ratio and the number of terms.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
This problem is not possible.

Here's why. The formula for the nth term of a geometric sequence is:
a%5Bn%5D+=+a%5B1%5D%2Ar%5E%28%28n-1%29%29
with a%5B1%5D being the first term, "r" being the common ratio, and "n" being the number of the term. So 2 and 2048 should fit this formula:
2048+=+2%2Ar%5E%28%28n-1%29%29
If we realize that 2048 = 2*2*2*2*2*2*2*2*2*2*2 then we know that the only factors of 2048 are 2's or powers of 2. 2's and powers of 2 cannot add up to an odd number like 273.

If we don't know this about 2048 we could try to go further. Dividing both sides by 2 we get:
1024+=+r%5E%28%28n-1%29%29
Next we will use the formula for the sum of a geometric series:
S%5Bn%5D+=+a%5B1%5D%28%281-r%5En%29%2F%281-r%29%29
where S%5Bn%5D is the sum of the first n terms, a%5B1%5D is the first term, "r" is the common ratio, and "n" is the number of terms. Substituting in our values for the sum and the first term we get:
273+=+2%28%281-r%5En%29%2F%281-r%29%29
Multiplying each side by (1-r) [to eliminate the fraction] we get:
273%281-r%29+=+2%281-r%5En%29
which simplifies to:
273-273r+=+2%281-r%5En%29
Next we return to our earlier equation:
1024+=+r%5E%28%28n-1%29%29
If we multiply each side of this by r we get:
1024%2Ar+=+r%5E%28%28n-1%29%29+%2Ar
On the right side we use the rule for exponents when multiplying (i.e. add the exponents:
1024%2Ar+=+r%5E%28%28n-1%29%29+%2Ar%5E1
1024r+=+r%5E%28%28n-1%29%2B1%29
1024r+=+r%5En
This gives us an expression to use back in:
273-273r+=+2%281-r%5En%29
Substituting in 1024r:
273-273r+=+2%281-1024r%29
Now we can solve for r. Multiplying out the right side:
273-273r+=+2-2048r
Adding 2048r:
273%2B1775r+=+2
Subtracting 273:
1775r+=+-271
Dividing by 1775:
r+=+%28-271%29%2F1775
With this value for "r", it will be impossible for the series to start at 2 and eventually get to 2048. (Try it and see!).

Since the formulas for geometric series do not work with a first term of 2, a last term of 2048 and a sum of 273, these numbers cannot reflect a geometric series.