Find the sum of the geometric series
48 + 120 + . . + 1875
a. 3093
b. 7780.5
c. 1218
d. 24037.5

  a. 3093

Step-by-step explanation:

The missing two terms in the 5-term sequence are ... 300 + 750.

You can add up these 5 terms directly, or you can recognize that the sum will be more than the last term (not C), but cannot be more than double the last term (not B or D). The sum must be 3093, choice A.

If you use the formula for the general term of the series, you can find the number of terms.

The common ratio is 120/48 = 2.5

The general term is ...

  an = a1·r^(n -1)

Solving for n, we get ...

  n = log(an/a1)/log(r) +1 = log(1875/48)/log(2.5) +1 = 5 . . . . the number of terms

Then the sum is given by ...

  Sn = a1(r^n -1)/(r -1) . . . . . using a1=48, r=2.5, n=5

  S5 = 48(2.5^5 -1)/(2.5 -1) = 48(96.65625/1.5)

  S5 = 3093

The sum of a geometric sequence with a common ratio of 2 is double the last term, less the first term. When the common ratio gets larger, the sum gets smaller than double the last term.

  Sn = a1(r^n -1)/(r -1) = (a1·r^n -a1)/(r -1) = (r·an -a1)/(r -1)

  Sn = (r/(r -1))an -a1/(r -1) . . . . . . verifies the above comment

In our case, this evaluates to ...

  S5 = (2.5/1.5)(1875) -48/1.5 = (5/3)(1875) -(2/3)(48)

  = 3125 -32 = 3093

Using the first and last terms this way, we only need the common ratio and don't need to know the number of terms.

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step-by-step explanation: