Question

The value of $\sqrt{42+\sqrt{42+\sqrt{42+ \cdots \infty}}}$ is
A. 7
B. 6
C. 5
D. 3

Answer 1

Hint: Assuming the given expression to be equal to some variable and then by squaring both the sides and then simplifying we can easily find its value.

Complete step-by-step answer:
We have to find the value of $\sqrt{42+\sqrt{42+\sqrt{42+\cdots \infty}}}$.
Let required value be x. Then,
$\sqrt{42+\sqrt{42+\sqrt{42+ \cdots \infty}}}=x$
Squaring both sides, we get,
$42+\sqrt{42+\sqrt{42+\cdots \infty}}=x^2$
Rearranging the terms, we get,
$\sqrt{42+\sqrt{42+\cdots \infty}}=x^2-42$
Now, as $\sqrt{42+\sqrt{42+ \cdots \infty}}=x$, thus, we get,
$x=x^2-42$
$\implies x^2-x-42=0$
Now we solve this quadratic equation formed by method of factorisation.
$x^2-x-42=0$ can be written as:
$x^2-7x+6x-42=0$
$\implies x(x-7)+6(x-7)=0$
$\implies (x-7)(x+6)=0$
$\implies x = 7 \ or \ -6$
As $x=\sqrt{42+\sqrt{42+ \cdots \infty}}$, hence, it cannot be negative.
Thus, x = 7
Hence, option A is correct.

Note: In this type of questions involving repeating square roots, we equate the expression to some variable and then squaring both sides and forming the quadratic equation, we can easily find the value of the expression.