Prove that Xn is decreasing?

Question

Xn +1 = sq. root(Xn+6)
Xsub1 = 4

Answer 1

X(n+1)^2 = X(n) + 6

X(n+1)^2 - X(n)^2 = Xn - Xn^2 + 6 = (2 - Xn)(Xn +3)

If Xn > 3, then X(n+1)^2 < X(n)^2, so we need to prove that Xn > 3.
Induction:
X(n+1) = sqrt(Xn + 6) > sqrt 9 = 3

Answer 2

Xn +1 = sq. root(Xn+6)
lim (sqrt(n+6)/(n+1) =
n --> infinity
lim (1/sqrt(n+6)/1= 0
n --> infinity

Since this limit is <1, the series converges by the ratio test.

Answer 3

it's decreasing. just take my word for it.