Please help solve this Math question!?

Well, simplifying this expression can be done in two ways, depending upon how much you already know. Since it seems that you have not gone very far, yet, it may be best to do it from first principles, and work it from the bottom up.

I write the expression like this because of the limitations of this format :

1 + 1/1+ .... 1/1+ ....1/1+ .... 1/x
........(a) ...... (b) ....... (c) ...... (d)

From (c) and (d) we get 1/(1 + 1/x), which = 1/( (x + 1)/x ) = x/(x + 1)

This makes (b) and {(c) and (d)} into 1/ { 1 + x/(x + 1) }

Inside the bracket we have { (x + 1) + x } / (x + 1) = (2x + 1) / (x + 1)

Hence, 1/ { 1 + x/(x + 1) } = (x + 1) / (2x + 1)

We now have (a) + { (b) and (c) and (d) } becomes 1 / { 1 + (x + 1) / (2x + 1) }

Inside the bracket we get { (2x + 1) + (x + 1) } / (2x+ 1) = (3x + 2) / (2x + 1)

Therefore 1 / { 1 + (x + 1) / (2x + 1) } = (2x + 1) / (3x + 2)

Now we just have to bring in the initial 1, and the whole expression becomes

1 + (2x + 1) / (3x + 2) = { (3x + 2) + (2x + 1) } / (3x + 2)

= (5x + 3) / (3x + 2)

which is as far as we can go, since we do not know what the expression is equal to, and we do not know what x is.

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The second way to do this is to start from the top end and evaluate the successive "convergents" to the fraction :

The first convergent is simply 1.

The second convergent is 1 + 1/1 = 2

The third convergent is 1 + 1/ ( 1 + 1/1) = 1 + 1/2 = 3/2

The fourth convergent is (3 + 2) / (2 + 1) = 5/3

The last convergent is (5x + 3) / (3x + 2), as before.

(I have not gone into details about how you get the successive convergents because it takes too long. If you have not yet learnt this, you probably will come to it soon.)

1 + 1/{1 + 1/[1 + 1/(1 + x)]}

x = 1 + 1/{1 + 1/[1 + 1/(1 + ...)]}

1 + 1/x = x
x(1 + 1/x) = x(x)
x + 1 = x^2
x^2 - x = 1
x^2 - x/2 - x/2 = 1
x^2 - x/2 - x/2 + 1/4 = 1 + 1/4
(x^2 - x/2) - (x/2 - 1/4) = 4/4 + 1/4
x(x - 1/2) - 1/2(x - 1/2) = 5/4
(x - 1/2)(x - 1/2) = 5/4
(x - 1/2)^2 = 5/4
x - 1/2 = ±√(5/4)
x = 1/2 ±(√5)/2
x = (1 ±√5)/2