Evaluate the Summation sum from k=5 to 20 of 26+4k

Problem

(∑_k=5^20)(26+4*k)

Solution

1. Identify the type of sequence being summed. The expression 26 + 4ki*s(l)*i*n*e*a*r*i*n*t*e*r*m*s(o)*ƒ$, which means this is an arithmetic series.

2. Determine the number of terms n in the summation. The sum starts at k=5 and ends at k=20

n=20−5+1=16

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

(a_1)=26+4*(5)=46

4. Calculate the last term (a_n) by substituting the upper limit k=20 into the expression.

(a_n)=26+4*(20)=106

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

(S_16)=(16*(46+106))/2

6. Simplify the expression to find the final value.

(S_16)=8*(152)=1216

Final Answer

(∑_k=5^20)(26+4*k)=1216

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