If m is any positive integer then the possible value of #sqrt(m+sqrt(m+sqrt(m+...)))-sqrt(m-sqrt(m-sqrt(m-...)))# is?

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Answer 1 · 2018-08-04T21:48:40

#1#

Let:

#u = sqrt(m+sqrt(m+sqrt(m+...)))#

#v = sqrt(m-sqrt(m-sqrt(m-...)))#

Then:

#u^2 = m+u " " => " " u^2-u-m = 0 " " => " " u = 1/2+1/2sqrt(4m+1)#

#v^2 = m-v " " => " " v^2+v-m = 0 " " => " " v = -1/2+1/2sqrt(4m+1)#

(noting that in each case we have to choose the positive root)

So:

#u - v = (1/2+1/2sqrt(4m+1))-(-1/2+1/2sqrt(4m+1))#

#color(white)(u-v) = 1#