Evaluate the Summation sum from K=1 to 7 of -3(2)^(k-1)

Problem

(∑_k=1^7)(−)*3*(2)^(k−1)

Solution

1. Identify the type of series. The expression −3*(2)^(k−1) is in the form a*(r)^(k−1) which represents a geometric series.

2. Determine the parameters of the sum. The first term a is found by substituting k=1 a=−3*(2)^(1−1)=−3 The common ratio r is 2 The number of terms n is 7

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

4. Substitute the values into the formula: (S_7)=(−3*(1−2^7))/(1−2)

5. Simplify the expression. Calculate the power: 2^7=128

6. Calculate the numerator and denominator: (S_7)=(−3*(1−128))/(−1)

7. Perform the final arithmetic: (S_7)=(−3*(−127))/(−1)=381/(−1)=−381

Final Answer

(∑_k=1^7)(−)*3*(2)^(k−1)=−381

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