Evaluate the Summation sum from i=1 to 10 of i(i^2+1)

Problem

(∑_i=1^10)(i)*(i2+1)

Solution

1. Distribute the term i inside the parentheses to rewrite the summand.

(∑_i=1^10)(i3+i)

2. Split the summation into two separate sums using the linearity property of summations.

(∑_i=1^10)(i3)+(∑_i=1^10)(i)

3. Apply the formula for the sum of the first n cubes, which is (∑_i=1^n)(i3)=((n*(n+1))/2)2 where n=10

((10*(11))/2)2=55=3025

4. Apply the formula for the sum of the first n integers, which is (∑_i=1^n)(i)=(n*(n+1))/2 where n=10

(10*(11))/2=55

5. Add the two results together to find the total sum.

3025+55=3080

Final Answer

(∑_i=1^10)(i)*(i2+1)=3080

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