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Let f: R → R be a function such that f(x + y) = f(x) + f(y) for every real x and y, and f(1) = 7. Then the value of ∑_(r=1)ⁿ f(r) is

1. (7n(n+1))/(2)
2. (7n)/(2)
3. (7(n+1))/(2)
4. 7n + (n+1)

Correct answer: (7n(n+1))/(2)

Solution

The additive (Cauchy) relation with f(1)=7 gives f(r)=7r. Then sum_{r=1}^n f(r) = 7(1+2+...+n) = 7n(n+1)/2.

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