Final answer:
To calculate the sum of the first 7 terms of the geometric sequence given, one must determine the common ratio, plug it into the sum formula for geometric series alongside the first term and the number of terms, and then calculate to obtain the sum rounded to the nearest hundredth.
Step-by-step explanation:
To find the sum of the first 7 terms of the given sequence 6, -10, 50/3, we first identify this as a geometric sequence by finding the common ratio (r). The common ratio is the factor that we multiply by to get from one term to the next. To find the ratio, we divide the second term by the first term:
r = -10 / 6 = -5 / 3
Now that we have the common ratio, we can use the formula for the sum of the first n terms of a geometric series:
sn = a1(1 - rn) / (1 - r)
For the sum of the first 7 terms, n = 7, a1 = 6, and r = -5/3:
s7 = 6(1 - (-5/3)7) / (1 - (-5/3))
Plugging in the values and simplifying, we get:
s7 = 6(1 - (-78125/2187)) / (1 + 5/3)
s7 = 6(2187 + 78125) / 2187) / (8/3)
s7 = 6(802312 / 2187) / (8/3)
s7 = (401156/1093.5) / (8/3)
Calculate this expression with a calculator to get the sum to the nearest hundredth.