Evaluate the Summation sum from i=1 to 8 of 2^(i-1)

Problem

(∑_i=1^8)(2(i−1))

Solution

1. Identify the type of series. The expression 2(i−1) represents a geometric series where the first term a is found by substituting i=1

a=2(1−1)=2=1

2. Determine the common ratio r by looking at the base of the exponential term.

r=2

3. Identify the number of terms n in the summation, which is the upper limit of the sum.

n=8

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

(S_8)=(1*(1−2))/(1−2)

5. Simplify the expression by calculating the power and performing the subtraction.

(S_8)=(1−256)/(−1)

6. Calculate the final value.

(S_8)=(−255)/(−1)=255

Final Answer

(∑_i=1^8)(2(i−1))=255

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