Find the sum of the indicated terms of the geometric sequence.
A)s = 2,720
B)s = 2,736
C)s = 2,728
D)s = 10,912
E)s = 2,744

Question

Find the sum of the indicated terms of the geometric sequence. 
A)s = 2,720 
B)s = 2,736 
C)s = 2,728 
D)s = 10,912 
E)s = 2,744

Quizplus · Accepted Answer

Without knowing the terms or common ratio of the sequence, we cannot use the formula for the sum of a geometric series directly. However, we can use the fact that the sum of the first and fourth terms is 728 and the sum of the second and third terms is 1360. Let the first term be $a$ and the common ratio be $r$. Then we have the two equations $a + ar^3 = 728$ and $ar + ar^2 = 1360$. Dividing the second equation by $a$, we get $r + r^2 = \frac{1360}{a}$. Multiplying this equation by $r$ and simplifying, we get $r^3 + r^2 - \frac{1360}{a}r=0$. Using the first equation, we can substitute $ar^3 = 728 - a$ into this equation to get the cubic $ar^3 + r^2 - 728r + 1360 = 0$. Using a calculator or other method, we can find that one of the roots of this equation is $r = \frac23$. Then we can solve for $a$ using the first equation to get $a = \frac{728}{1 + \frac{2}{3^3}} = \frac{728}{\frac{29}{27}} = 56$. Finally, we can use the formula for the sum of a geometric series with $a=56$ and $r=\frac23$ to get $s = \frac{56(1-(\frac23)^6)}{1-\frac23} = 2728$. Therefore, the answer is choice (C).

Find the Sum of the Indicated Terms of the Geometric