Evaluate sigma-summation Underscript i equals 1 Overscript 9 End Scripts negative 28 left parenthesis negative one half right parenthesis superscript i minus 1 baseline using the finite geometric sum formula. Round your answer to three decimal places.
To evaluate the given series using the finite geometric sum formula, we need to first identify the common ratio (r) and the first term (a).
In this case, the common ratio can be found by dividing any term by the previous term. Let's divide the second term by the first term:
-28(-1/2) / -28 = 1/2 = r
So, we have a common ratio of 1/2.
The first term (a) can be found by plugging in the value of i=1 into the given expression:
a = (-28) (-1/2)^(1-1) = -28
Now, we can use the formula for the sum of a finite geometric series:
S = a (1 - r^n) / (1 - r)
where S is the sum, a is the first term, r is the common ratio, and n is the number of terms.
In this case, n = 9 (as given in the summation notation).
Plugging in the values, we have:
S = (-28) (1 - (1/2)^9) / (1 - 1/2)
Calculating the expression inside the parentheses:
S = (-28) (1 - 1/512) / (1/2)
Simplifying:
S = (-28) (511/512) / (1/2)
S = -28 * (511/512) * 2
S = -28 * (511/256)
S = -1428.875 (rounded to three decimal places)
Therefore, the sum of the given series is approximately -1428.875.