Question

Evaluate √20+√20+√20.............

Answer 1

Given,

An expression √20+√20+√20.............

To Find,

The value of √20+√20+√20.............

Solution,

Let us assume the value of the given expression be x

Then

x= √20+( √20+( √ 20...

x= √(20+x)

squaring both sides,

x^2 =20 +x

x^2 -x-20=0

x^2-5x+4x-20=0

x (x-5)+4 (x-5 )=0

(x+4)(x-5)=0

x = -4 or 5

The value of x cannot be negative as under root can have a negative value.

Hence, the value of √20+√20+√20............. is 5.

Answer 2

Let us take given term as $x$.

$x=\sqrt{20\sqrt{20\sqrt{20\sqrt{20.......} } } }$

This can be written as:

$x=\sqrt{20+x}$

Now, squaring on both side, we get:

$x^{2} =20+x$

$x^{2} -x-20=0$

Now, by splitting the middle term, we get:

$x^{2} -5x+4x-20=0$

$x(x-5)+4(x-5)=0$

$(x+4)(x-5)=0$

$x=-4$ or $x=5.$

We know $x$ cannot be negative.

Therefore:

$x=5.$