Evaluate Using Summation Formulas sum from i=1 to 11 of 5*2^i

Problem

(∑_i=1^11)(5)⋅2

Solution

1. Identify the type of series. The expression 5⋅2 represents a geometric series where each term is multiplied by a common ratio r=2

2. Determine the parameters of the sum. The first term a is found by substituting i=1 which gives a=5⋅2=10 The common ratio is r=2 and the number of terms is n=11

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

4. Substitute the values into the formula:

(S_11)=(10*(2−1))/(2−1)

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

(S_11)=(10*(2048−1))/1

6. Calculate the final product:

(S_11)=10⋅2047=20470

Final Answer

(∑_i=1^11)(5)⋅2=20470

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