Evaluate the Summation sum from k=1 to 6 of 6k^3

Problem

(∑_k=1^6)(6)*k3

Solution

1. Factor out the constant 6 from the summation to simplify the calculation.

(∑_k=1^6)(6)*k3=6*(∑_k=1^6)(k3)

2. Identify the formula for the sum of the first n cubes, which is (∑_k=1^n)(k3)=((n*(n+1))/2)2

(∑_k=1^n)(k3)=((n*(n+1))/2)2

3. Substitute n=6 into the formula to find the sum of the cubes from 1 to 6

(∑_k=1^6)(k3)=((6*(6+1))/2)2

4. Simplify the expression inside the parentheses.

(6*(7))/2=42/2=21

5. Square the result to find the value of the inner summation.

21=441

6. Multiply the result by the constant 6 that was factored out earlier.

6×441=2646

Final Answer

(∑_k=1^6)(6)*k3=2646

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