KCET · Maths · Mathematical Induction
If ' \(n\) ' is a positive integer, then \(\mathrm{n}^{3}+2 \mathrm{n}\) is divisible by
- A 2
- B 6
- C 15
- D 3
Answer & Solution
Correct Answer
(D) 3
Step-by-step Solution
Detailed explanation
Let \(P(n)=n^{3}+2 n\)
\[
\begin{array}{lc}
\Rightarrow & \mathrm{P}(1)=1+2=3 \\
\Rightarrow & \mathrm{P}(2)=8+4=12 \\
\Rightarrow & \mathrm{P}(3)=27+6=33
\end{array}
\]
Here, we see that all these numbers are divisible by 3 .
\[
\begin{array}{lc}
\Rightarrow & \mathrm{P}(1)=1+2=3 \\
\Rightarrow & \mathrm{P}(2)=8+4=12 \\
\Rightarrow & \mathrm{P}(3)=27+6=33
\end{array}
\]
Here, we see that all these numbers are divisible by 3 .
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