# Thread: Square route question

1. ## Square route question

Find the pattern or general rule between $\sqrt{2*3*4*5+1}$ and $\sqrt{0*1*2*3+1}$ and $\sqrt{3*4*5*6+1}$

I can't see any pattern between them, let alone working out a general rule. Can somebody give me a hint, not the full answer, just a flavour please

2. Originally Posted by Mukilab
Find the pattern or general rule between $\sqrt{2*3*4*5+1}$ and $\sqrt{0*1*2*3+1}$ and $\sqrt{3*4*5*6+1}$

I can't see any pattern between them, let alone working out a general rule. Can somebody give me a hint, not the full answer, just a flavour please
List them out...

3. Hello, Mukilab!

Find the pattern or general rule for: . $\begin{Bmatrix} \sqrt{0\!\cdot\!1\!\cdot\!2\!\cdot\!3+1} \\ \sqrt{1\!\cdot\!2\!\cdot\!3\!\cdot\!4+1} \\ \sqrt{2\!\cdot\!3\!\cdot\!4\!\cdot\!5+1} \\ \sqrt{3\!\cdot\!4\!\cdot\!5\!\cdot\!6+1} \end{Bmatrix}$

We have:

. . $\begin{array}{ccccccc}\sqrt{0\!\cdot\!1\!\cdot\!2\ !\cdot\!3 + 1} &=^& \sqrt{1} &=& 1 \\
\sqrt{1\!\cdot\!2\!\cdot\!3\!\cdot\!4+1} &=& \sqrt{25} &=& 5 \\
\sqrt{2\!\cdot\!3\!\cdot\!4\!\cdot\!5+1} &=& \sqrt{121} &=& 11 \\
\sqrt{3\!\cdot\!4\!\cdot\!5\!\cdot\!6+1} &=& \sqrt{361} &=& 19 \\
\sqrt{4\!\cdot\!5\!\cdot\!6\!\cdot\!67 + 1} &=& \sqrt{841} &=& 29 \\ \vdots && \vdots \end{array}$

The general term is:

. . $a_n \;=\;\sqrt{n(n+1)(n+2)(n+3)+1}\;\text{ for } n = 0,1,2,\hdots$

. . . . . $=\; \sqrt{n^4 + 6n^3 + 11n^2 + 6n + 1}$

. . . . . $=\;\sqrt{(n^2 + 3n + 1)^2}$

. . . . . $=\; n^2+3n+1$