MHB Deferred Periodicity of a Number

  • Thread starter Thread starter kaliprasad
  • Start date Start date
kaliprasad
Gold Member
MHB
Messages
1,333
Reaction score
0
A decimal number is said to be deferred periodicity if it is periodic preceded by one or more digits

show that $\frac{1}{n} + \frac{1}{n+1}+ \frac{1}{n+2}$ forms a decimal fraction of deferred periodicity
 
Mathematics news on Phys.org
My solution:

At least one of the 3 denominators must be divisible by 2, and so will be a terminating decimal, and at least one of the three denominators must be divisible by 3, and so will be a repeating decimal, and so the sum of the three fractions will be of deferred periodicity regardless of the periodicity of the fraction whose denominator may not be divisible by either 2 or 3.
 
The answer by MARKFL is right

my solution

above sum is

$\dfrac{3n^2+ 6n + 2 }{n(n+1)(n+2)}$

numerator is not divisible by 3 but denominator is divisible

so it is periodic as 3 is not a factor of 10

again if n id odd then numerator odd but denominator is even so 2 divides denominator but not numerator so it is deferred

if n is even $3n^2+6n$ is divisible by 4 but 2 is not so numerator is not divisible by 4 but denominator is divisible by 4 (actually 8) so in reduced for numerator is odd and denominator is even so if is deferred
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Is it possible to arrange six pencils such that each one touches the other five? If so, how? This is an adaption of a Martin Gardner puzzle only I changed it from cigarettes to pencils and left out the clues because PF folks don’t need clues. From the book “My Best Mathematical and Logic Puzzles”. Dover, 1994.
Back
Top