Evaluate the Summation sum from i=1 to 8 of 7(7/10)^(i-1)

Problem

(∑_i=1^8)(7)*(7/10)(i−1)

Solution

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

2. Determine the parameters of the series. The first term is a=7 the common ratio is r=7/10 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)=(7*(1−(7/10)8))/(1−7/10)

5. Simplify the denominator:

1−7/10=3/10

6. Divide the first term by the denominator:

7/3/10=70/3

7. Calculate the final expression:

(S_8)=70/3*(1−(7/10)8)

8. Evaluate the power and the final numerical value:

(7/10)8=5764801/100000000=0.05764801

(S_8)=70/3*(1−0.05764801)

(S_8)=70/3*(0.94235199)

(S_8)=21.9882131

Final Answer

(∑_i=1^8)(7)*(7/10)(i−1)=659646393/30000000

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