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Find the limit of the infinite sum of this function

Source: MTRP 2016 Class 11-Multiple Choice Question: Problem 2 :-

May 31, 2020
functionlimitSummationinfinite sum

Problem Statement

Let ff be a function satisfying f(x+y+z)=f(x)+f(y)+f(z)f(x+y+z)=f(x)+f(y)+f(z) for all integers xx, yy, zz. Suppose f(1)=1f(1)=1, f(2)=2f(2)=2. Then lim⁡n→∞1n3∑r=1n4rf(3r)\lim \limits_{n\to \infty} \frac{1}{n^3} \sum \limits_{r=1}^n 4rf(3r) equals
[*] 4 [*] 6 [*] 12 [*] 24
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