Evaluate the Summation sum from k=10 to 60 of 2k

Problem

(∑_k=10^60)(2)*k

Solution

1. Factor out the constant 2 from the summation to simplify the expression.

(∑_k=10^60)(2)*k=2*(∑_k=10^60)(k)

2. Rewrite the summation as the difference of two sums starting from k=1 to use the standard arithmetic series formula.

(∑_k=10^60)(k)=(∑_k=1^60)(k)−(∑_k=1^9)(k)

3. Apply the formula for the sum of the first n integers, which is (∑_k=1^n)(k)=(n*(n+1))/2

(∑_k=1^60)(k)=(60*(60+1))/2

(∑_k=1^9)(k)=(9*(9+1))/2

4. Calculate the values of the two sums.

(60⋅61)/2=30⋅61=1830

(9⋅10)/2=9⋅5=45

5. Subtract the smaller sum from the larger sum to find the value of the series from k=10 to 60

1830−45=1785

6. Multiply by the constant 2 that was factored out in the first step.

2⋅1785=3570

Final Answer

(∑_k=10^60)(2)*k=3570

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