Question

The value of $\sqrt {72 + \sqrt {72 + \sqrt {72 + .........} } } $ is

Answer 1

Hint:
We will let the expression be $x$. Then, take square on both sides. Substitute the value of the expression again and form a quadratic equation. Factorise the equation formed and equate each factor to 0 to find the value of the given expression.

Complete step by step solution:
We have to find the value of $\sqrt {72 + \sqrt {72 + \sqrt {72 + .........} } } $
Let the value of $\sqrt {72 + \sqrt {72 + \sqrt {72 + .........} } } $ be $x$
Then, taking square both sides,
${x^2} = 72 + \sqrt {72 + \sqrt {72 + \sqrt {72 + .........} } } $
Now, substitute the value of $\sqrt {72 + \sqrt {72 + \sqrt {72 + .........} } } $ , that is $x$
$
  {x^2} = 72 + x \\
   \Rightarrow {x^2} - x - 72 = 0 \\ $
We will factorise the above equation.
$
   \Rightarrow {x^2} - 9x + 8x - 72 = 0 \\
   \Rightarrow x\left( {x - 9} \right) + 8\left( {x - 9} \right) = 0 \\
   \Rightarrow \left( {x - 9} \right)\left( {x + 8} \right) = 0 \\
$
We will equate each factor to 0 to find the value of $x$
$x = 9, - 8$
But, the square root of any number cannot be negative.

Hence, the value of $\sqrt {72 + \sqrt {72 + \sqrt {72 + .........} } } $ is 9.

Note:
There are infinite terms in a given expression. Even if we take one term out by squaring it, the number of terms in square root will not change. Also, we have to discard negative value as the square root of any number cannot be negative.