AP EAMCET · Maths · Functions
If \(\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}\) is defined by \(\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}(\mathrm{x})+\mathrm{f}(\mathrm{y}) \forall \mathrm{x}, \mathrm{y} \in\) \(\mathbb{R}\) and \(f(1)=7\), then \(\sum_{r=1}^n f(r)=\)
- A \(\frac{3 n(n+2)}{4}\)
- B \(\frac{\mathrm{n}(\mathrm{n}-1)}{2}\)
- C \(\frac{7 n(n+1)}{2}\)
- D \(\frac{(\mathrm{n}+1)(\mathrm{n}+2)}{4}\)
Answer & Solution
Correct Answer
(C) \(\frac{7 n(n+1)}{2}\)
Step-by-step Solution
Detailed explanation
Since \(f(x+y)=f(x)+f(y), \forall x, y \in R\) \(\Rightarrow f(x)=a x\) where \(a \in R\) Since given, \(\mathrm{f}(1)=7 \Rightarrow 7=\mathrm{a} \times 1 \Rightarrow \mathrm{a}=7\) \(\Rightarrow \mathrm{f}(\mathrm{x})=7 \mathrm{x}\) Now,…
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