Evaluate the Summation sum from i=6 to 12 of -5i+17

Problem

(∑_i=6^12)(−5*i+17)

Solution

1. Identify the type of sequence being summed. The expression −5*i+17 is linear in i which means this is an arithmetic series.

2. Determine the number of terms n The sum goes from i=6 to i=12 so n=12−6+1=7

3. Calculate the first term (a_1) by substituting the lower limit i=6 into the expression.

(a_1)=−5*(6)+17=−30+17=−13

4. Calculate the last term (a_n) by substituting the upper limit i=12 into the expression.

(a_n)=−5*(12)+17=−60+17=−43

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

(S_7)=(7*(−13+(−43)))/2

6. Simplify the expression to find the final value.

(S_7)=(7*(−56))/2=7*(−28)=−196

Final Answer

(∑_i=6^12)(−5*i+17)=−196

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