Evaluate the Summation sum from i=1 to 7 of 2^(i-1)

Problem

(∑_i=1^7)(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 and the number of terms n The common ratio is the base of the exponent, and the number of terms is the upper limit of the summation.

r=2

n=7

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

(S_7)=(1*(2−1))/(2−1)

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

2=128

(S_7)=(128−1)/1

(S_7)=127

Final Answer

(∑_i=1^7)(2(i−1))=127

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