Evaluate the Summation sum from i=1 to 5 of (2/3)^(i+1)

Problem

(∑_i=1^5)(2/3)

Solution

1. Identify the type of series. This is a finite geometric series with the general term (a_i)=(2/3)(i+1)

2. Determine the first term a Substitute i=1 into the expression:

a=(2/3)(1+1)=(2/3)2=4/9

3. Identify the common ratio r The ratio between consecutive terms is:

r=2/3

4. Identify the number of terms n Since the index i goes from 1 to 5 there are:

n=5

5. Apply the formula for the sum of a finite geometric series (S_n)=(a*(1−rn))/(1−r)

(S_5)=(4/9*(1−(2/3)5))/(1−2/3)

6. Simplify the denominator:

1−2/3=1/3

7. Simplify the expression by dividing the first term by the denominator:

(S_5)=4/9⋅3⋅(1−32/243)

(S_5)=4/3⋅(1−32/243)

8. Calculate the final value:

(S_5)=4/3⋅211/243

(S_5)=844/729

Final Answer

(∑_i=1^5)(2/3)=844/729

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