AP EAMCET · Maths · Functions
\(f: \mathbb{R} \rightarrow \mathbb{R}\) is defined by \(f(x+y)=f(x)+12 y, \forall x, y \in \mathbb{R}\) If \(f(1)=6\), then \(\sum_{r=1}^n f(r)=\)
- A \(n^2\)
- B \(5 n^2\)
- C \(6 n^2\)
- D \(\frac{3 n(n+1)}{2}\)
Answer & Solution
Correct Answer
(C) \(6 n^2\)
Step-by-step Solution
Detailed explanation
\(f^{\prime}(x)=12\), Given \(f(1)=6\) \(f(2)=f(1)+12(1)=6+12=18\) Similarly, \(f(3)=30 ; f(4)=42\) \(\begin{aligned} & \sum_{r=1}^n f(r)=6+18+30+42 \ldots n \text { terms } \\ & =6(1+3+5+7 \ldots . n \text { terms })=6 . n^2 \end{aligned}\)
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