If f:R to R satisfies f(x + y) = f(x) + f(y) , for all x,y ∈ R and f(1) = 7 , then Σlimits_{r = 1

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1.Relation and Function

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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

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