Q on sqrts and sequences: sqrt(2 + sqrt(4 + ...))
- Thread starter tomas_xc
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It's not really striking me how to SIMPLIFY the expression. If it were infinite, it would be easier.
If
\(\displaystyle \sqrt{2+\sqrt{4+\sqrt{8+...}}} = x\)
then
\(\displaystyle \sqrt{2+\sqrt{4+2\sqrt{2+\sqrt{4+\sqrt{8+...}}+...}}} = x\)
and
\(\displaystyle \sqrt{2+\sqrt{4+2x}} = x\)
That's not too hard to solve for the four Real solutions. Obviously, only the positive one would be useful.
Perhaps it will lead ot something for your finite case.
If
\(\displaystyle \sqrt{2+\sqrt{4+\sqrt{8+...}}} = x\)
then
\(\displaystyle \sqrt{2+\sqrt{4+2\sqrt{2+\sqrt{4+\sqrt{8+...}}+...}}} = x\)
and
\(\displaystyle \sqrt{2+\sqrt{4+2x}} = x\)
That's not too hard to solve for the four Real solutions. Obviously, only the positive one would be useful.
Perhaps it will lead ot something for your finite case.