SOLUTION: Let z1, z2 and z3 be three complex numbers in geometric progression. Suppose that the average of these numbers is 10, while the average of their squares is 20i. Determine the val

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: Let z1, z2 and z3 be three complex numbers in geometric progression. Suppose that the average of these numbers is 10, while the average of their squares is 20i. Determine the val      Log On


   

Question 383150: Let z1, z2 and z3 be three complex numbers in geometric progression. Suppose that the
average of these numbers is 10, while the average of their squares is 20i. Determine the
value of z2, the middle term.
Found 2 solutions by Jk22, robertb:
Answer by Jk22(389) About Me  (Show Source):
You can put this solution on YOUR website!
Geometric progression means : z2=a*z1, z3=a^2*z1

then

sum : z1%2A%28a%5E3-1%29%2F%28a-1%29=30
sum of square :

the first sum squared is : z1%5E2%2A%28a%5E3-1%29%5E2%2F%28%28a-1%29%5E2%29=900
divided by the sum of squared :

%28a%5E3-1%29%5E2%2F%28a-1%29%5E2%2A%28a%5E2-1%29%2F%28a%5E6-1%29=-20%2F3%2Ai


%28a%5E3-1%29%2F%28a-1%29%2A%28a%2B1%29%2F%28a%5E3%2B1%29=-20%2F3%2Ai| use (a^3+1)=(a+1)(a^2-a+1)

%28a%5E2-a%2B1%29%2F%28a%5E2%2Ba%2B1%29=-20%2F3%2Ai
a%5E2-a%2B1=-20%2F3%2Ai%2A%28a%5E2%2Ba%2B1%29
%281%2B20%2F3%2Ai%29a%5E2%2B%28-1%2B20%2F3%2Ai%29a%2B%281%2B20%2F3%2Ai%29=0

Let D+=%28-1%2B20%2F3%2Ai%29%5E2-4%2A%281%2B20%2F3%2Ai%29%5E2

then a=%28-%28-1%2B20%2F3%2Ai%29%2B-sqrt%28D%29%29%2F%282%281%2B20%2F3%2Ai%29%29
and z1=300%2F%28a%5E2%2Ba%2B1%29

Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
let z1 = a
z2 = ar, and
z3+=+ar%5E2. Then from the given,
%28a%2Bar%2Bar%5E2%29%2F3+=+10, and %28a%5E2%2Ba%5E2r%5E2+%2B+a%5E2r%5E4%29%2F3+=+20i, both arising from the given.
Hence
a%2Bar%2Bar%5E2+=+30, (1) and
a%5E2%281%2Br%5E2+%2B+r%5E4%29+=+60i. (2)
Divide (2) by (1), to get
%28a%5E2%281+%2B+r%5E2+%2B+r%5E4%29%29%2F%28a%281%2Br%2Br%5E2%29%29+=+%2860i%29%2F%2830%29, or
a%281-r+%2B+r%5E2%29+=+2i, or a-ar+%2B+ar%5E2+=+2i. Subtract this result from
a%2Bar%2Bar%5E2+=+30, to get
2ar+=+30+-+2i, OR, ar+=+15+-+i. BUT ar is exactly the middle term of the geometric sequence of 3 geometric terms, or z2.
Thus, z2 = 15 - i.