Evaluate the Summation sum from n=1 to 8 of -4(1/3)^(n-1)

Problem

(∑_n=1^8)(−)*4*(1/3)(n−1)

Solution

1. Identify the type of series. This is a finite geometric series with the general form (∑_n=1^k)(a)*r(n−1)

2. Determine the parameters of the series. The first term is a=−4 the common ratio is r=1/3 and the number of terms is k=8

3. Apply the formula for the sum of a finite geometric series, which is (S_k)=(a*(1−rk))/(1−r)

4. Substitute the values into the formula:

(S_8)=(−4*(1−(1/3)8))/(1−1/3)

5. Simplify the denominator:

1−1/3=2/3

6. Calculate the power in the numerator:

(1/3)8=1/6561

7. Simplify the expression further:

(S_8)=(−4*(1−1/6561))/2/3

8. Multiply by the reciprocal of the denominator:

(S_8)=−4⋅3/2⋅6560/6561

9. Reduce the fraction:

(S_8)=−6⋅6560/6561

10. Divide by the common factor of 3:

(S_8)=−2⋅6560/2187

11. Multiply to find the final value:

(S_8)=−13120/2187

Final Answer

(∑_n=1^8)(−)*4*(1/3)(n−1)=−13120/2187

Want more problems? Check here!