Question

here's a question highlighted in the above picture

Answer 1

answer is three, short cut to the problem, since it's √(6+(√6.....) factors being 2 and 3, difference is 1, since the sign is a positive one, the answer is the bigger number that is 3. had it been a negative sign the answer would be 2. only works when factors differ by 1.

Answer 2

Hey There!!!


This question is an interesting one with terms continuing in a pattern till infinity.

Let us solve it.
Let the given value be x. That is:

$x=\sqrt{6+\sqrt{6+\sqrt{6+...}}}$

We can see that Value of x must be positive.


Now, when we see, this is an infinite part. We can write the inside term as x, as follows:


$x=\sqrt{6+x}$


Now, we can solve it easily:

$x=\sqrt{6+x} \\ \\ \implies x^2=6+x \\ \\ \implies x^2-x-6=0 \\ \\ \implies x^2-3x+2x-6=0 \\ \\ \implies x(x-3)+2(x-3)=0 \\ \\ \implies (x-3)(x+2)=0 \\ \\ \implies x=-2 \, \, OR \, \, x=3$


But since we have concluded above that Value of x is postive, we have:

$\boxed{x=3}$

Thus, Answer is 3.


Hope it helps
Purva
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