Question

$\left\{n(n+1)(2n+1):n\in Z\right\}$is a subset of

• A. $6K:K\in Z$
• B. $12K:K\in Z$
• C. $18K:K\in Z$
• D. $24K:K\in Z$

Answer 1

The correct option is A

$6K:K\in Z$

Explanation for the correct option

Finding $\left\{n(n+1)(2n+1):n\in Z\right\}$is a subset of

Let $S\left(n\right)=\left\{n(n+1)(2n+1)/n\in Z\right\}$

Put $n=1$

$$\begin{gathered} S\left(1\right)=\left\{1(1+1)(2\times 1+1)\right\} \\ S\left(1\right)=\left\{2\left(3\right)\right\} \\ S\left(1\right)=6 \end{gathered}$$

Similarly put $n=2$

$$\begin{gathered} S\left(2\right)=\left\{2(2+1)(2\times 2+1)\right\} \\ S\left(2\right)=\left\{2\times 3\times 5\right\} \\ S\left(2\right)=6\times 5 \end{gathered}$$

Its is clear from the above examples that $\left\{n(n+1)(2n+1):n\in Z\right\}$is a subset of $6K:K\in Z$

Hence the correct answer is option (A).