(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]
A) 1.5 (and almost: 2.99999999.....i.e., writing 3 as this so its integer part would be 2)
B) phi = (sqrt(5)1)/2
C) 1/phi^2
D) sqrt(2)/2  1 and sqrt(2)/2 + 1
A graphing calculator, or a computer program that produces graphs like one, is useful in finding between which integers the numbers lie, so as to know [x] and to formulate x  [x]. The graphing led to the pseudosolution part of A. Plot y=x and also y = 2*int(x)  (xint(x)) for the first example (A), and see where the broken curve crosses the straight line.
Once the integer portion is found so that it can be put explicitly in the equations, an arithmetic progression A,B,C is solved for C via 2BA=C, with the appropriate substitution of, in the above case, R for C:
2(1)(P1)=P
2P=3
P=1.5
Similarly for geometric progression A,B,C, B^2 / A = C:
2(1^2)/(Q1)=Q to get Q = phi
and with the harmonic progression, A,B,C, 2/B  1/A = 1/C:
2/(1)  1/(S+1) = 1/S
and
2/1  1/(s1) = 1/S.

Posted by Charlie
on 20080322 15:17:21 