Look at this:
x + x + x + x + x + x + x + x + . . . = y
y + y + y + y + y + y + y + y + . . . = 5
Can you find x + y ?
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I think Karthik Sharma's solution might not be clear 2 every one. Here's an elaborated solution of the above. Consider the equation [(sqrt{x + sqrt{x + sqrt{x + sqrt{x + sqrt{x + sqrt{x + sqrt{x + sqrt{x + sqrt{x + .............) = y] be equation 1. Add x on both sides, we get[x + sqrt{x + sqrt{x + sqrt{x + sqrt{x + sqrt{x + sqrt{x + sqrt{x + sqrt{x + sqrt{x + .........] = x + y. Now Square root on both sides. We get [sqrt{x + sqrt{x + sqrt{x + sqrt{x + sqrt{x + sqrt{x + sqrt{x + sqrt{x + sqrt{x + sqrt{x + ...........] = sqrt{x + y}. The term on the Left hand side of the equation is the same as y. So sqrt{x + y} = y. Similarly, consider the equation [(sqrt{y + sqrt{y + sqrt{y + sqrt{y + sqrt{y + sqrt{y + sqrt{y + sqrt{y + sqrt{y + ...........) = 5] be equation 2 and perform the above steps for the equation 2. We get sqrt{y + 5} = 5. On solving, we get y = 20, which implies sqrt{x + y} = 20. Isn't that Simple?
very halpfull thanks
Can you tell me more how can we write the 1st equation as x + y = y ?
That term remains same if you remove one x from it because it has infinite terms. So if you add one x or remove one x it doesn't make any differnce
Substitute "y" value in LHS. ;)
Here is my simple and clear solution for this problem. Let x=\sqrt{y+\sqrt{y+\ldots}}. This strongly suggests that x^2 = 25 = y + \sqrt{y+\sqrt{y+\ldots}} = y + x. This further implies that 5 = \sqrt{25} = \sqrt{y + x}.
did you just take the expression to a nearest value ,, i mean is it accurate ? also if you please tell me how to know more about number theory ?
there are Infinite terms, So there will be no difference if you do that for solving
Not so understand.. how first equation and second equation being generated?
Here is my simple and clear solution for this problem. Let x=\sqrt{y+\sqrt{y+\ldots}}. This strongly suggests that x^2 = 25 = y + \sqrt{y+\sqrt{y+\ldots}} = y + x. This further implies that 5 = \sqrt{25} = \sqrt{y + x}.
lol , i just did it in 20 sec. very simple
i did same\
nice concept
Nice!!!!!!
Great one!
I'm sorry but I beg to disagree with Karthik Sharma's solution. His rewritten equations seem improper and unjustified. Here is my simple and clear solution for this problem. Let x=\sqrt{y+\sqrt{y+\ldots}}. This strongly suggests that x^2 = 25 = y + \sqrt{y+\sqrt{y+\ldots}} = y + x. This further implies that 5 = \sqrt{25} = \sqrt{y + x}.
cant u write in systematic manner so that we all can understand?? Using real symbols?
I wanted not to write much. And the solution is correct as we are adding infinite terms
look , in diffrentiation we write infite funtions y=x^1/2^x^1/2.... so on as a funtion with x=x^1/2y as it goes on infitinatniously which is our function ! so thts how we get (x+y)1/2=y! not a trick actualyy higher math
Karthik's solution strongly suggest that y= 20. However, x = 5. So his solution is still wrong.
Let us first find the value of y . We can do that by squaring the second equation, getting:
y + y + y + … = 5 ( 2 )
y + y + y + y + … = 2 5 ⇒ y + y + y + … = 2 5 − y ⇒
⇒ 5 = 2 5 − y ⇒ y = 2 0 .
Similarly, we can now acquire the value of x as well:
x + x + x + … = 2 0 ( 2 )
x + x + x + x + … = 4 0 0 ⇒ x + x + x + … = 4 0 0 − x ⇒
⇒ 2 0 = 4 0 0 − x ⇒ x = 3 8 0 .
From here, we can clearly see that x + y = 4 0 0 = 2 0 .
what I don't get is how can you get whole numbers when you square root y infinitely (I assume this because it states (sqrt{y + sqrt{y + sqrt{y + sqrt{y + sqrt{y + sqrt{y + sqrt{y + sqrt{y + sqrt{y + ...........))
Last time I checked infinity is not a solution to anything, that's the problem that physicists faced with trying to combine quantum mechanics and the general theory of relativity. Maybe I'm wrong in the way I'm looking at this, maybe in math this acceptable.
If there wasn't an infinite amount of each x's and y's, (meaning if both equations had an end point) I would just keep eliminating the square roots by increasing both sides by the power of 2 until I eliminated all of the square roots to find the value of y then I would solve for x in the same method.
In my opinion, this is something totally acceptable in mathematics. The thing is, infinity is more of an abstract term, rather than something we can feel/touch/see in practical problems. The thing is, in this problem (although it might seem a bit ambiguous at first) there is indeed an infinite amount of x 's and y 's. Technically speaking, the given sum CAN be a whole number (and it is for the given value of y ). Same goes for x .
Unfortunately, I'm really bad at physics to be able to discuss this any further. However you do hold a valid point yourself!
hi,if you do that ,which have many step to do and by accident, you make it more complex
I'm totally with your opinion which is relating the mathematical form with our nature, I think we must neglect such forms as -in my opinion- it won't help us in comprehensing our real world.
it's not clear
Which part of the solution in particular is not clear to you?
its a infinite series question even if u remove one x and one y from beginning the expressions remain same,,,,,,, do urself after this hint
square the second equation yields y + 5 = 25, therefore y = 20.
square the first equation yields x + y = y^2 or x + 20 = 400, hence x = 380.
finally sqrt(x+y) = sqrt(380+20) = sqrt(400) = 20,
This is an approach from middle school. more sophisticated solution can be found from Karthik Sharma :)
Square both sides of the second equation.
y + sqrt( y + sqrt( y + sqrt( ....))) = 25
=> y + 5 = 25
=> y = 20
Substitute y = 20 in first equation. Square both sides.
x + sqrt( x + sqrt( x + sqrt( ....))) = 400
=> x + y = 400
=> sqrt(x + y) = 20
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The 1st equation can be written as -
x + y = y
And, the Second Equation as -
y + 5 = 5
squaring both the sides, we get
y + 5 = 2 5 So, y = 2 0
Which means,
x + y = 2 0