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

Answer:   a. 3093Step-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 = 2.5The 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 termsThen 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 commentIn our case, this evaluates to ...   S5 = (2.5/1.5)(1875) -48/1.5 = (5/3)(1875) -(2/3)(48)   = 3125 -32 = 3093Using the first and last terms this way, we only need the common ratio and don't need to know the number of terms.