Evaluate the Summation sum from n=1 to 20 of 2n+1

Problem

(∑_n=1^20)(2*n+1)

Solution

1. Identify the type of sequence being summed. The expression 2*n+1 is a linear function of n which indicates an arithmetic series.

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

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

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

(a_20)=2*(20)+1=41

4. Apply the formula for the sum of an arithmetic series, which is (S_k)=(k*((a_1)+(a_k)))/2 where k is the number of terms.

(S_20)=(20*(3+41))/2

5. Simplify the expression to find the total sum.

(S_20)=10*(44)=440

Final Answer

(∑_n=1^20)(2*n+1)=440

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