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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 function f is additive, meaning it can be expressed in the form f(x) = cx for some constant c. Given that f(1) = 7, we find c = 7, leading to f(x) = 7x. The sum A3_(r=1)ⁿ f(r) becomes A3_(r=1)ⁿ 7r, which simplifies to 7(n(n+1)/2) using the formula for the sum of the first n integers.

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