limit of sqrs

[math_Lover]

What is the limit of [MATH]\sqrt6,\sqrt{6+\sqrt6},\sqrt{6+\sqrt{6+\sqrt6}},\cdots[/MATH]I would appreciate an answer that includes both a way and a final answer.

Thanks

 

[Deleted member 4993]

What is the limit of [MATH]\sqrt6,\sqrt{6+\sqrt6},\sqrt{6+\sqrt{6+\sqrt6}},\cdots[/MATH]I would appreciate an answer that includes both a way and a final answer.

Thanks

a₁ = √6, a₂ =√(6+a₁), a₃ =√(6+a₂) ...... a_n = ?

Please show us what you have tried and exactly where you are stuck.

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Please share your work/thoughts about this problem.

 

[Romsek]

consider the sequence of terms produced by the rule.

[MATH]a_{n+1} = \sqrt{6 + a_n},~a_1=\sqrt{6}[/MATH]
Now in the limit what can you say about [MATH]a_n[/MATH] and [MATH]a_{n+1}[/MATH] ?

 

[math_Lover]

Converges to something like 3?

 

[tkhunny]

What kind of determination is "something like". Please do better than that.

If $$x = \sqrt{6+\sqrt{6+\sqrt{6+...}}}$$,
Then $$x = \sqrt{6+x}$$
Please show YOUR work in the future.

 

[Steven G]

I repeat, x = sqrt(6+x)

 

[pka]

What is the limit of [MATH]\sqrt6,\sqrt{6+\sqrt6},\sqrt{6+\sqrt{6+\sqrt6}},\cdots[/MATH]I would appreciate an answer that includes both a way and a final answer.

Suppose $\ell=\sqrt6,\sqrt{6+\sqrt6},\sqrt{6+\sqrt{6+\sqrt6}},\cdots$ then $\ell=\sqrt{6+\ell~~}$
So that $\ell^2-\ell+6=0$ or $(\ell-3)(\ell+2)=0$ Which of values is the answer and explain why.

 

[Deleted member 4993]

Converges to something like 3?

Correct

 

[skeeter]

Converges to something like 3?

easy when someone else tells you ...

limit of sqrs

What is the limit of \sqrt6,\sqrt{6+\sqrt6},\sqrt{6+\sqrt{6+\sqrt6}},\cdots I would appreciate an answer that includes both a way and a final answer. Thanks :)

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