Find \(\sum_{t=1}^{39} f(t)\) if \(f: \mathbf{R} \rightarrow \mathbf{R}\) is defined as \(f(x+y)=f(x)+f(y),…
Find \(\sum_{t=1}^{39} f(t)\) if \(f: \mathbf{R} \rightarrow \mathbf{R}\) is defined as \(f(x+y)=f(x)+f(y), x, y \in \mathbf{R}\) and \(f(\mathrm{I})=7\)
5187
5460
5740
5407
Solution
It is given that, \(f(x+y)=f(x)+f(y)\) and
\(\begin{aligned}
& f(1)=7 \\
& f(x)=7 x
\end{aligned}\)
\(\therefore \quad f(x)=7 x\)
So,
\(\begin{aligned}
\sum_{t=1}^{39} f(t) & =7[1+2+3+\ldots+39] \\
& =7 \times \frac{39 \times 40}{2}=5460.
\end{aligned}\)