Evaluate the Summation sum from k=1 to 8 of 4(2/3)^(k-1)

Problem

(∑_k=1^8)(4)*(2/3)(k−1)

Solution

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

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

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

4. Substitute the values into the formula:

(S_8)=(4*(1−(2/3)8))/(1−2/3)

5. Simplify the denominator:

1−2/3=1/3

6. Calculate the expression by multiplying by the reciprocal of the denominator:

(S_8)=4⋅3⋅(1−256/6561)

7. Perform the final arithmetic:

(S_8)=12⋅6305/6561

8. Reduce the fraction by dividing both 12 and 6561 by their greatest common divisor, 3:

(S_8)=4⋅6305/2187

(S_8)=25220/2187

Final Answer

(∑_k=1^8)(4)*(2/3)(k−1)=25220/2187

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