(Solution 1)We have: \(\sqrt{1+1\cdot2\cdot3\cdot4}=1+1\cdot4=5\)

\(\sqrt{1+2\cdot3\cdot4\cdot5}=1+2\cdot5=11\)

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\(\sqrt{1+204\cdot205\cdot206\cdot207}=1+204\cdot207=42229\)

(Solution 2) We have: \(\sqrt{1+1\cdot2\cdot3\cdot4}=2\cdot3-1=5\)

\(\sqrt{1+2\cdot3\cdot4\cdot5}=3\cdot4-1=11\)

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\(\sqrt{1+204\cdot205\cdot206\cdot207}=205\cdot206-1=42229\)

Alone has done solutions 3+4.

(Solution 1)We have: 1+1⋅2⋅3⋅4−−−−−−−−−−−√=1+1⋅4=5

1+2⋅3⋅4⋅5−−−−−−−−−−−√=1+2⋅5=11

.........

1+204⋅205⋅206⋅207−−−−−−−−−−−−−−−−−−√=1+204⋅207=42229

(Solution 2) We have: 1+1⋅2⋅3⋅4−−−−−−−−−−−√=2⋅3−1=5

1+2⋅3⋅4⋅5−−−−−−−−−−−√=3⋅4−1=11

..............

1+204⋅205⋅206⋅207−−−−−−−−−−−−−−−−−−√=205⋅206−1=42229

Alone has done solutions 3+4.

Solution 1:1+204.205.206.207−−−−−−−−−−−−−−−−√=1783288441−−−−−−−−−√=42229

Solution 2:1+n(n+1)(n+2)(n+3)−−−−−−−−−−−−−−−−−−−−−−√=1+(n2+3n)(n2+3n+2)−−−−−−−−−−−−−−−−−−−−−−√=(n2+3n)2+2.(n2+3n)+1−−−−−−−−−−−−−−−−−−−−−−−√

=(n2+3n+1)2−−−−−−−−−−−−√=n2+3n+1

With n=204 then 1+204.205.206.207−−−−−−−−−−−−−−−−√=2042+3.204+1=42229

Solution 1:\(\sqrt{1+204.205.206.207}=\sqrt{1783288441}=42229\)

Solution 2:\(\sqrt{1+n\left(n+1\right)\left(n+2\right)\left(n+3\right)}=\sqrt{1+\left(n^2+3n\right)\left(n^2+3n+2\right)}=\sqrt{\left(n^2+3n\right)^2+2.\left(n^2+3n\right)+1}\)

\(=\sqrt{\left(n^2+3n+1\right)^2}=n^2+3n+1\)

With n=204 then \(\sqrt{1+204.205.206.207}=204^2+3.204+1=42229\)