Question

If $n\in N$, then $n^3+2n$ is divisible by

• A. $2$
• B. $3$
• C. $5$
• D. $6$

Answer 1

Answer: (B)

$3$

$f\left(n\right)=n^3+2n$
put $n=1$, to obtain $f\left(1\right)=1^3+2.1=3$
Therefore, $f\left(1\right)$ is divisible by $3$
Assume that for $n=k$, $f\left(k\right)=k^3+2k$ is divisible by $3$
Now, $f(k+1)={(k+1)}^3+2(k+1)=k^3+2k+3(k^2+k+1)=f\left(k\right)+3(k^2+k+1)$
Since, $f\left(k\right)$ is divisible by $3$
Therefore, $f(k+1)$ is divisible by $3$
and from the principle of mathematical induction $f\left(n\right)$ is divisible by $3$ for all $n\in N$
Ans: B