Evaluate the Summation sum from n=1 to 14 of 3n+2

Problem

(∑_n=1^14)(3*n+2)

Solution

1. Identify the type of sequence being summed. The expression 3*n+2 is linear in n which means this is an arithmetic series.

2. Find the first term (a_1) by substituting n=1 into the expression.

(a_1)=3*(1)+2=5

3. Find the last term (a_14) by substituting n=14 into the expression.

(a_14)=3*(14)+2=44

4. Determine the number of terms k in the summation.

k=14−1+1=14

5. Apply the formula for the sum of an arithmetic series, (S_k)=(k*((a_1)+(a_k)))/2

(S_14)=(14*(5+44))/2

6. Simplify the expression to find the total sum.

(S_14)=7*(49)=343

Final Answer

(∑_n=1^14)(3*n+2)=343

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