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.

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) (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:

with 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:

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:

Next we will use the formula for the sum of a geometric series:

where is the sum of the first n terms, 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:

Multiplying each side by (1-r) [to eliminate the fraction] we get:

which simplifies to:

Next we return to our earlier equation:

If we multiply each side of this by r we get:

On the right side we use the rule for exponents when multiplying (i.e. add the exponents:

This gives us an expression to use back in:

Substituting in 1024r:

Now we can solve for r. Multiplying out the right side:

Adding 2048r:

Subtracting 273:

Dividing by 1775:

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.
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