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1.Relation and Function
medium
If $f:R \to R$ satisfies $f(x + y) = f(x) + f(y)$, for all $x,\;y \in R$ and $f(1) = 7$, then $\sum\limits_{r = 1}^n {f(r)} $ is
A
$\frac{{7n}}{2}$
B
$\frac{{7(n + 1)}}{2}$
C
$7n(n + 1)$
D
$\frac{{7n(n + 1)}}{2}$
(AIEEE-2003)
Solution
(d) $f(x + y) = f(x) + f(y)$
Put $x = 1,\,y = 0$==> $f(1) = f(1) + f(0) = 7$
Put $x = 1,\,y = 1$ ==> $f(2) = 2.f(1) = 2.7$
Similarly $f(3) = 3.7$ and so on
$\therefore \sum\limits_{r = 1}^n {f(r) = 7\,(1 + 2 + 3 + ….. + n)} $
$= \frac{{7n(n + 1)}}{2}$.
Standard 12
Mathematics