(A) Determine all possible non zero real P such that {P}, [P] and P are in arithmetic sequence.
(B) Determine all possible non zero real Q such that {Q}, [Q] and Q are in geometric sequence.
(C) Determine all possible non zero real R such that [R], {R} and R are in geometric sequence.
(D) Determine all possible non zero real S such that {S}, [S] and S are in
harmonic sequence.
Note: [x] is defined as the greatest integer ≤ x and {x} = x - [x]
Used in all cases: P = [P]+{P}
(A) 2*[P]={P}+P
=> 2*P-2*{P}={P}+P
=> P=3*{P}
=> [P]=2*{P} => 2*{p} must be an integer and {P}<1
This is possible only if {P}=0.5 or 0.
=> P=1.5 is the only possible non zero case possible.
(B) [Q]²={Q}Q => (Q/{Q})²-3Q/{Q}+1=0
Let Q/{Q}=x
=> x=(3+√5)/2 or (3-√5)/2
So, Q=(3+√5)/2*{Q} or (3-√5)/2*{Q}
=> [Q]²={Q}²*(3+√5)/2 or (3-√5)/2*{Q}²
=> [Q]= {Q}*(1+√5)/2 or {Q}*(√5-1)/2
If [Q]!=0, then |[Q]/{Q}| > 1
=> [Q]={Q}*(1+√5)/2
=> {Q}=(-1+√5)/2
=> Q = (1+√5)/2 = 1.618033988...
(C) {R}²=R*[R]
R=(3+√5)/2*[R] or (3-√5)/2*[R]
But if [R]>0, then 1 ≤|R/[R]| < 2
both of the equations dont satisfy this condition.
If [R]=0 => R=0.
If [R] ≤ -1, 0 < |R/[R]| ≤ 1
R=(3-√5)/2*[R]
=> {R}²=(3-√5)/2 => {R}=(√5-1)/2 => R=-(3-√5)/2 = -0.3819660
(D) 2/[S]=1/{S}+1/S
=> (S+{S})*(S-{S}) = 2*S*{S}
=> (S/{S})²-2*S/{S}-1=0
=> S = (√2 +1)*{S} or (1-√2)*{S}
=> [S]=√2*{S} or -√2*{S}
=> 0≤{S}<1 => 0 ≤√2*{S} < √2 or -√2 ≤-√2*{S} < 0
[S] can take 1 or -1 as its values as it is an integer
=> {S}=1/√2
=> S = 1+1/√2 or -1+1/√2 = 1.707106 or -0.292893Edited on March 24, 2008, 4:01 am
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Posted by Praneeth
on 2008-03-24 03:39:36 |
Full Solution
