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

Problem

(∑_n=1^8)(5)*r^(n−1)

Solution

1. Identify the type of series. The expression 5*r^(n−1) represents a geometric series where the first term a is found by substituting n=1

a=5*r^(1−1)=5*r^0=5

2. Determine the common ratio and the number of terms. The common ratio is r and the summation runs from n=1 to n=8 so the number of terms k is 8

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

(S_8)=(5*(1−r^8))/(1−r)

Final Answer

(∑_n=1^8)(5)*r^(n−1)=(5*(1−r^8))/(1−r)

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