Question

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)} $.
A $\dfrac{{7n}}{2}$
B $\dfrac{{7\left( {n + 1} \right)}}{2}$
C $7n(n + 1)$
D $\dfrac{{7n(n + 1)}}{2}$

Answer 1

Hint: Given, $f:R \to R$ satisfies $f(x + y) = f(x) + f(y)$ for all $x,y \in R$ and $f(1) = 7$. We have to find the value of $\sum\limits_{r = 1}^n {f(r)} $. We will find the value of $f(2),\,f(3),\,f(4),......$ and so on. Then we will generalize a formula for $f(n)$. Then, we will use $1 + 2 + 3 + 4 + .......n = \dfrac{{n(n + 1)}}{2}$ to find the value of $\sum\limits_{r = 1}^n {f(r)} $.

Formula Used: $1 + 2 + 3 + 4 + .......n = \dfrac{{n(n + 1)}}{2}$

Complete step by step solution:A relation between a collection of inputs and outputs is known as a function. A function is, to put it simply, a relationship between inputs in which each input is connected to precisely one output. Each function has a range, codomain, and domain. The usual way to refer to a function is as$f(x)$, where $x$ is the input. A function is typically represented as $y = f(x)$. In mathematics, a function is a unique arrangement of the inputs (also referred to as the domain) and their outputs (sometimes referred to as the codomain), where each input has exactly one output and the output can be linked to its input.
 Given, $f:R \to R$ satisfies $f(x + y) = f(x) + f(y)$ for all $x,y \in R$ and $f(1) = 7$.
Given, $f(1) = 7$
$f(2) = f(1 + 1)$
Using $f(x + y) = f(x) + f(y)$
$ = f(1) + f(1)$
Putting value of $f(1)$
$ = 7 + 7$
$ = 14$
$f(3) = f(2 + 1)$
Using $f(x + y) = f(x) + f(y)$
$ = f(2) + f(1)$
Putting value of $f(1)$ and $f(2)$
$ = 14 + 7$
$ = 21$
Similarly, $f(4) = f(3 + 1)$
Using $f(x + y) = f(x) + f(y)$
$ = f(3) + f(1)$
Putting value of $f(1)$ and $f(3)$
$ = 21 + 7$
$ = 28$
And so on
Generalizing the formula for n
$f(n) = 7 + (n - 1)7$
$f(n) = 7 + 7n - 7 = 7n$
Now, $\sum\limits_{r = 1}^n {f(r)} = f(1) + f(2) + f(3) + f(4) + ........ + f(n)$
$ = 7 + 14 + 21 + 28.......7n$
Taking 7 common
$ = 7(1 + 2 + 3 + 4 + ........ + n)$
We know that $1 + 2 + 3 + 4 + .......n = \dfrac{{n(n + 1)}}{2}$
$ = 7\left( {\dfrac{{n\left( {n + 1} \right)}}{2}} \right)$
After simplifying above expression
$ = \dfrac{{7n\left( {n + 1} \right)}}{2}$

Hence, option D is correct.

Note: Students should know how to correctly use given information for solving the question. They should use $f(x + y) = f(x) + f(y)$ this given relation correctly in order to generate a generalized formula for n. And use the formula of sum of the first natural number to get the required answer.