Verified answer

3 = sqrt(x + sqrt(x + sqrt(x + sqrt(x + .....

Square both sides

9 = x + sqrt(x + sqrt(x + sqrt(x + .....

9 = x + 3

6 = x

EDIT:

It's just a substitution

We already stated that 3 = sqrt(x + sqrt(x + sqrt(x + ....

So we can rewrite that as 3 = sqrt(x + 3)

Remember, this sequence continues on forever. If it stops at any time, the substitution can't work.

Here's how I approach these kind of infinite series:

I think you meant to denote like this: √(x+√(x+√(x + ... = 3, rather than closing it off at x.

So lets say the ... stuff in the + ...) has a value of U.

So then we have √(x+√(x+√(x + U))) = 3.

Still looks complex, but here's where a little imagination is necessary. Remember that it went on forever, before, so the √(x+√(x+√(x + U))) has to equal √(x+√(x+U)). Notice that I removed one of the parenthesis expressions and replaced it with U again.

Because remember that U has a value of the ... stuff so we could expand it out 5 times before changing to U and it should still equal 3, because those expressions go on forever. It's like when you put 2 mirrors facing each other and it looks like it goes to infinity, but each one contributes a smaller and smaller portion (that we can tell).

So now we have 2 equations: U = √(x + U), and √(x + U) = 3.

Lets take the first one and square both sides: U² = x + U, do the same for the other: x + U = 9.

So x = 9 - U, and plug this into the quadratic: U² = 9 - U + U = 9. So U = +3 or U = -3.

So when U = 3 then x = 9 - 3 = 6. when U = -3, then x = 9 - -3 = 12.

Now here's the issue: we said that U = √(x + U), but if we use U = -3, then we will have -3 = 3, so we have to stick with U=3 and x = 6, which is what you had.

To see how this works, I built a spreadsheet that shows how fast this converges to 3. I'll upload it to my Google docs so you can see it. If you have Excel or OpenOffice you can make one.

In cell D2 type this: =SQRT(B$2+D1), then in cell B2 put 6. That is the value of x in this formula.

Now copy the cell D2 down to D3, D4, so that the formula will reference the cell above it, and cell B2 (because of the dollar sign in B$2). As long as you have 6 in B2 and something greater than -6 in D1 (that's what I call the 'seed' number or my guess for the value of U), then it converges to 3 rapidly.

The fact that it is infinite number of times makes the equation simpler to solve! If it is 3 times deep for example, you would need to expand the equations to find x.

With infinite times, the LHS can be simplified as below:

√(x+√(x+√x) ) … = y

√(x+y) = y

Now, since y =3, when both sides power by 2:

x + 3 = 9, therefore x = 6.

Ahh, the good old nested radical, at least that's what I assume you've meant here. If so, you've put the parentheses in the wrong spot.

√(x + √(x + √(x + ... ))) = 3

But anyway, notice that because this is infinite the left hand side contains itself on the first level, so we have

√(x + 3) = 3

Now this is easily solvable with simple algebra.

x + 3 = 9

x = 6

Put 6 levels of this in your calculator and you'll get 2.999928... and some more decimals. Clearly this is converging to 3.

sqrt(x+sqrtx+sqrtx + -------------------- = 3

sqrt (x + 3 ) = 3

or x+ 3 = 9

x= 9-3

x= 6 answer

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proof

sqrt(6+sqrt6+------------ = sqrt(6+3) = 3 = R.H.S.