Weird nested sequence! $V_{ij}$

Number Theory Level 4

Consider the sequence $3,18,243,...n×3^{2^{n-1}}$

If this sequence is enclosed in nested square roots upto infinite terms i.e. if

$\sqrt{3+\sqrt{18+\sqrt{243+...}}}=k$ ,

Find the value of $\left\lceil{k}\right\rceil$ .

Note:

This problem is similar to the below problem.

Nested Radicals

The answer is 4.

1 solution

Michael Mendrin
Jun 18, 2014

Okay, I lost a lot of time on this one before realizing that there is no explicit expression for this value. It's

$\sqrt { 3 } V=\sqrt { 3 } (1.757932756680045327...)=3.044828850751997370...$

where $V$ is the Vijayaraghavan’s inﬁnite nested radical constant, the infinite nested radical being

$V=\sqrt { 1+\sqrt { 2+\sqrt { 3+\sqrt { 4+\sqrt { 5+... } } } } }$

so that the answer here would be $4$

It's not hard to see how to come up with the nested radical as given in this problem from this.

The best strategy for this problem would be finding upper and lower bounds. Namely, a lower bound > 4 and an upper bound < 5.

Michael Tong - 6 years, 12 months ago

Well, Michael, if one uses a calculator, it doesn't take long to find out that it's got to be between 3 and 4. I was just curious to know if there was an exact expression for it.

Michael Mendrin - 6 years, 11 months ago

Oh shit

Didn't know that this sequence of V is already found and was feeling happier that this was first put forth by me.

Vinay Sipani - 6 years, 11 months ago

please elabourate your solution

Dheeraj Agarwal - 6 years, 8 months ago

Weird nested sequence! V i j V_{ij} V i j ​

Note:

The answer is 4.

1 solution

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Weird nested sequence! $V_{ij}$