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

Problem

(∑_n=1^8)(5)*(1/5)n

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 first term (a_1) Substitute n=1 into the expression:

(a_1)=5*(1/5)1=1

3. Identify the common ratio r and the number of terms k The common ratio is r=1/5 and the number of terms is k=8

4. Apply the formula for the sum of a finite geometric series (S_k)=((a_1)*(1−rk))/(1−r)

(S_8)=(1*(1−(1/5)8))/(1−1/5)

5. Simplify the denominator.

1−1/5=4/5

6. Calculate the final value by multiplying by the reciprocal of the denominator.

(S_8)=5/4*(1−1/5)

7. Evaluate the power 5

5=390625

8. Finalize the arithmetic.

(S_8)=5/4*(390624/390625)=97656/78125

Final Answer

(∑_n=1^8)(5)*(1/5)n=97656/78125

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