Suppose x is a positive real number such that x .[ x ] and x are in the geometric progression. Find the least positive integer n such that x^n>100 . (Here [x] denotes the integer part of x and x =x-[x])

Step-by-step solution

aligned & [x]^2=x x & x =a & [x]=a r & x=a r^2 & a+a r=a r^2 & r^2-r-1=0 & r= 1 5 2 r= 1+ 5 2 & a r=I & a= 2 I (1+ 5 ) & a= I( 5 -1) 2 aligned aligned & 0 100 N ₁₀ ( 5 +1 2 )>2 & N>9.5 N_ =10 aligned