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

Problem

(∑_n=1^7)(2)*(−2)(n−1)

Solution

1. Identify the type of series. This is a finite geometric series because each term is found by multiplying the previous term by a constant ratio.

2. Determine the parameters of the series. The first term (a_1) is found by substituting n=1

(a_1)=2*(−2)(1−1)=2*(−2)0=2

The common ratio r is the base of the exponent:

r=−2

The number of terms n is the upper limit of the summation:

n=7

3. Apply the formula for the sum of a finite geometric series (S_n)=((a_1)*(1−rn))/(1−r)

(S_7)=(2*(1−(−2)7))/(1−(−2))

4. Simplify the expression by calculating the power and the denominator:

(S_7)=(2*(1−(−128)))/(1+2)

(S_7)=(2*(1+128))/3

5. Calculate the final value:

(S_7)=(2*(129))/3

(S_7)=258/3

(S_7)=86

Final Answer

(∑_n=1^7)(2)*(−2)(n−1)=86

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