What is the result of the sum f(1)+...+f(20) if f(x)=x^2+2x+3

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We'll evaluate the sum by substituting x by the values 1,...,20.

f(1) = 1^2 + 2*1 + 3

f(2) = 2^2 + 2*2 + 3

f(3) = 3^2 + 2*3 + 3

......................

f(20) = 20^2 + 2*20 + 3

We'll calculate the sum:

f(1) + f(2) +..+ f(20) = (1^2+...+20^2) + 2*(1+...+20) + 3*20

We'll remove the brackets and we'll substitute the sum of the squares of the first 20 natural numbers, by the product:

1^2+...+20^2 = 20*(20+1)(2*20+1)/6

1^2+...+20^2 = 10*21*41/3

1^2+...+20^2 = 10*7*41

1^2+...+20^2 = 2870

We'll remove the brackets and we'll substitute the sum the first 20 natural numbers, by the product:

1+2+...+20 = 20(20+1)/2

1+2+...+20 = 10*21

1+2+...+20 = 210

f(1) + f(2) +..+ f(20) = 2870 + 2*210 + 3*20

**f(1) + f(2) +..+ f(20) = 3350**

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