Evaluate the Summation sum from i=1 to 20 of -3i+33

Problem

(∑_i=1^20)(−3*i+33)

Solution

1. Identify the type of series by observing that the expression −3*i+33 is linear in i which represents an arithmetic series.

2. Determine the first term (a_1) by substituting i=1 into the expression.

(a_1)=−3*(1)+33=30

3. Determine the last term (a_20) by substituting i=20 into the expression.

(a_20)=−3*(20)+33=−27

4. Identify the number of terms n in the summation, which is the upper limit minus the lower limit plus one.

n=20−1+1=20

5. Apply the formula for the sum of an arithmetic series, (S_n)=(n*((a_1)+(a_n)))/2

(S_20)=(20*(30+(−27)))/2

6. Simplify the expression to find the final sum.

(S_20)=10*(3)=30

Final Answer

(∑_i=1^20)(−3*i+33)=30

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