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Question Number 153866 by amin96 last updated on 11/Sep/21

3+(√(3+(√(6+(√(9+(√(12+…+(√(99))))))))))=?

$$\mathrm{3}+\sqrt{\mathrm{3}+\sqrt{\mathrm{6}+\sqrt{\mathrm{9}+\sqrt{\mathrm{12}+\ldots+\sqrt{\mathrm{99}}}}}}=? \\ $$

Answered by peter frank last updated on 11/Sep/21

152122

$$\mathrm{152122} \\ $$

Commented by amin96 last updated on 11/Sep/21

  how did you do the solution sir?

$$ \\ $$how did you do the solution sir?

Commented by mr W last updated on 11/Sep/21

i don′t think it can be solved in closed  form.  no matter how you got the result 152122,  the result is wrong.    S=3+(√(3+(√(6+(√(9+(√(....+(√(99))))))))))    <3+(√(99+(√(99+(√(99+(√(....+(√(99))))))))))     <3+(√(99+(√(99+(√(99+(√(....+(√(99+(√(...))))))))))))     =3+L  with L=(√(99+(√(99+(√(99+(√(....+(√(99+(√(...))))))))))))  L=(√(99+L))  L^2 −L−99=0  L=((1+(√(397)))/2)≈10.5  ⇒3+(√(3+(√(6+(√(9+(√(....+(√(99)))))))))) < 3+L=13.5  so it can never be 152122!

$${i}\:{don}'{t}\:{think}\:{it}\:{can}\:{be}\:{solved}\:{in}\:{closed} \\ $$$${form}. \\ $$$${no}\:{matter}\:{how}\:{you}\:{got}\:{the}\:{result}\:\mathrm{152122}, \\ $$$${the}\:{result}\:{is}\:{wrong}. \\ $$$$ \\ $$$${S}=\mathrm{3}+\sqrt{\mathrm{3}+\sqrt{\mathrm{6}+\sqrt{\mathrm{9}+\sqrt{....+\sqrt{\mathrm{99}}}}}} \\ $$$$\:\:<\mathrm{3}+\sqrt{\mathrm{99}+\sqrt{\mathrm{99}+\sqrt{\mathrm{99}+\sqrt{....+\sqrt{\mathrm{99}}}}}} \\ $$$$\:\:\:<\mathrm{3}+\sqrt{\mathrm{99}+\sqrt{\mathrm{99}+\sqrt{\mathrm{99}+\sqrt{....+\sqrt{\mathrm{99}+\sqrt{...}}}}}} \\ $$$$\:\:\:=\mathrm{3}+{L} \\ $$$${with}\:{L}=\sqrt{\mathrm{99}+\sqrt{\mathrm{99}+\sqrt{\mathrm{99}+\sqrt{....+\sqrt{\mathrm{99}+\sqrt{...}}}}}} \\ $$$${L}=\sqrt{\mathrm{99}+{L}} \\ $$$${L}^{\mathrm{2}} −{L}−\mathrm{99}=\mathrm{0} \\ $$$${L}=\frac{\mathrm{1}+\sqrt{\mathrm{397}}}{\mathrm{2}}\approx\mathrm{10}.\mathrm{5} \\ $$$$\Rightarrow\mathrm{3}+\sqrt{\mathrm{3}+\sqrt{\mathrm{6}+\sqrt{\mathrm{9}+\sqrt{....+\sqrt{\mathrm{99}}}}}}\:<\:\mathrm{3}+{L}=\mathrm{13}.\mathrm{5} \\ $$$${so}\:{it}\:{can}\:{never}\:{be}\:\mathrm{152122}! \\ $$

Commented by MJS_new last updated on 11/Sep/21

can we find a closed form if it doesn′t stop?  (√(1×3+(√(2×3+(√(3×3+...(√(n×3)))))))) with n→∞  or at least for  (√(1+(√(2+(√(3+(√(4+...(√n))))))))) with n→∞

$$\mathrm{can}\:\mathrm{we}\:\mathrm{find}\:\mathrm{a}\:\mathrm{closed}\:\mathrm{form}\:\mathrm{if}\:\mathrm{it}\:\mathrm{doesn}'\mathrm{t}\:\mathrm{stop}? \\ $$$$\sqrt{\mathrm{1}×\mathrm{3}+\sqrt{\mathrm{2}×\mathrm{3}+\sqrt{\mathrm{3}×\mathrm{3}+...\sqrt{{n}×\mathrm{3}}}}}\:\mathrm{with}\:{n}\rightarrow\infty \\ $$$$\mathrm{or}\:\mathrm{at}\:\mathrm{least}\:\mathrm{for} \\ $$$$\sqrt{\mathrm{1}+\sqrt{\mathrm{2}+\sqrt{\mathrm{3}+\sqrt{\mathrm{4}+...\sqrt{{n}}}}}}\:\mathrm{with}\:{n}\rightarrow\infty \\ $$

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