Evaluate the Summation sum from i=5 to 13 of 3i+2

Problem

(∑_i=5^13)(3*i+2)

Solution

1. Identify the type of sequence being summed. The expression 3*i+2 is linear, meaning this is an arithmetic series.

2. Determine the number of terms n in the summation. The index goes from i=5 to i=13

n=13−5+1

n=9

3. Calculate the first term (a_1) by substituting the lower limit i=5 into the expression.

(a_1)=3*(5)+2

(a_1)=17

4. Calculate the last term (a_n) by substituting the upper limit i=13 into the expression.

(a_n)=3*(13)+2

(a_n)=41

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

(S_9)=(9*(17+41))/2

6. Simplify the expression to find the final sum.

(S_9)=(9*(58))/2

(S_9)=9*(29)

(S_9)=261

Final Answer

(∑_i=5^13)(3*i+2)=261

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