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If f:R to R satisfies f(x+y)=f(x)+f(y),...

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_(r=1)^(n)f( r)` is

A

`(7n)/(2) `

B

`(7(n+1))/(2)`

C

`7n(n+1) `

D

`(7n(n+1))/(2)`

Text Solution

AI Generated Solution

To solve the problem, we start with the functional equation given: 1. **Functional Equation**: We have \( f(x+y) = f(x) + f(y) \) for all \( x, y \in \mathbb{R} \). 2. **Finding \( f(2) \)**: We know \( f(1) = 7 \). To find \( f(2) \), we can use the functional equation by letting \( x = 1 \) and \( y = 1 \): \[ f(2) = f(1 + 1) = f(1) + f(1) = 7 + 7 = 14. \] ...
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