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  4. If f: R → R satisfies f (x + y) = f (x) + f (y), for all x, y ∈ R and f (1) = 7, then displaystyle ∑r=1n f(r) is

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Q. If $f : R → R$ satisfies $f (x + y) = f (x) + f (y)$, for all $x, y ∈ R$ and $f (1) = 7$, then $\displaystyle \sum_{r=1}^n f(r)$ is

AIEEEAIEEE 2003Relations and Functions

Solution:

$f (1) = 7$
$f (1 + 1) = f (1) + f (1)$
$f (2) = 2 × 7$
only $f (3) = 3 × 7$
$\displaystyle \sum_{r=1}^n f(r) = 7 (1 + 2 + ……… + n)$
$ =7 \frac{n\left(n+1\right)}{2}.$