Evaluate the Summation sum from n=1 to 8 of (1/5)^(n-1)

Problem

(∑_n=1^8)(1/5)

Solution

1. Identify the type of series. This is a finite geometric series with the general form (∑_k=0^n−1)(a)*r^k

2. Determine the parameters of the series. The first term a is found by substituting n=1 which gives a=(1/5)^(1−1)=1 The common ratio is r=1/5 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−r^n))/(1−r)

4. Substitute the values into the formula:

(S_8)=(1*(1−(1/5)^8))/(1−1/5)

5. Simplify the denominator:

1−1/5=4/5

6. Calculate the power in the numerator:

(1/5)^8=1/390625

7. Perform the subtraction in the numerator:

1−1/390625=390624/390625

8. Divide the numerator by the denominator by multiplying by the reciprocal:

(S_8)=390624/390625⋅5/4

9. Simplify the resulting fraction:

(S_8)=97656/78125

Final Answer

(∑_n=1^8)(1/5)=97656/78125

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