Evaluate the Summation sum from n=0 to 8 of 1/(6^n)

Problem

(∑_n=0^8)(1/6)

Solution

1. Identify the type of series. This is a finite geometric series with the first term a=1/6=1 the common ratio r=1/6 and the number of terms k=9 (since it starts at n=0 and ends at n=8.

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

3. Substitute the values into the formula:

(S_9)=(1*(1−(1/6)9))/(1−1/6)

4. Simplify the denominator:

1−1/6=5/6

5. Multiply by the reciprocal of the denominator:

(S_9)=6/5*(1−1/6)

6. Distribute and calculate the final numerical value:

(S_9)=6/5−6/(5⋅6)

(S_9)=6/5−1/(5⋅6)

(S_9)=(6⋅6−1)/(5⋅6)

(S_9)=(6−1)/(5⋅6)

7. Evaluate the powers of 6:

6=1679616

6=10077696

8. Calculate the final fraction:

(S_9)=(10077696−1)/(5⋅1679616)

(S_9)=10077695/8398080

9. Reduce the fraction by dividing both numerator and denominator by 5:

(S_9)=2015539/1679616

Final Answer

(∑_n=0^8)(1/6)=2015539/1679616

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