Evaluate the Summation sum from i=1 to 4 of (-1/7)^i

Problem

(∑_i=1^4)(−1/7)

Solution

1. Identify the type of sum, which is a finite geometric series with the first term a=−1/7 common ratio r=−1/7 and number of terms n=4

2. Apply the formula for the sum of a finite geometric series (S_n)=(a*(1−rn))/(1−r)

3. Substitute the values into the formula:

(S_4)=(−1/7*(1−(−1/7)4))/(1−(−1/7))

4. Simplify the denominator:

1−(−1/7)=1+1/7=8/7

5. Simplify the numerator by calculating the power:

(−1/7)4=1/2401

6. Calculate the term inside the parentheses:

1−1/2401=2400/2401

7. Multiply by the first term a

−1/7⋅2400/2401=−2400/16807

8. Divide by the simplified denominator:

−2400/16807⋅7/8=−300/2401

Final Answer

(∑_i=1^4)(−1/7)=−300/2401

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