Evaluate the Summation sum from i=1 to 20 of (i-1)^2

Problem

(∑_i=1^20)(i−1)

Solution

1. Change the index of the summation to simplify the expression by letting k=i−1

2. Determine the new limits for the summation: when i=1 k=0 when i=20 k=19

3. Rewrite the sum using the new variable k

(∑_k=0^19)(k2)

4. Remove the zero term since 0=0 which does not change the value of the sum.

(∑_k=1^19)(k2)

5. Apply the power sum formula for the sum of the first n squares, which is (∑_k=1^n)(k2)=(n*(n+1)*(2*n+1))/6

6. Substitute n=19 into the formula.

(19*(19+1)*(2*(19)+1))/6

7. Simplify the arithmetic expression.

(19⋅20⋅39)/6

8. Calculate the final product.

14820/6=2470

Final Answer

(∑_i=1^20)(i−1)=2470

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