MHB Geometric Series with Complex Numbers

[Yankel]

Hello all,

Three consecutive elements of a geometric series are:

m-3i, 8+i, n+17i

where n and m are real numbers. I need to find n and m.

I have tried using the conjugate in order to find (8+i)/(m-3i) and (n+17i)/(8+i), and was hopeful that at the end I will be able to compare the real and imaginary parts of the ratio, but I got difficult algebraic expressions, so I figure out it's not the way. Can you assist please ?

Thanks !

 

[SuperSonic4]

Hello all,

Three consecutive elements of a geometric series are:

m-3i, 8+i, n+17i

where n and m are real numbers. I need to find n and m.

I have tried using the conjugate in order to find (8+i)/(m-3i) and (n+17i)/(8+i), and was hopeful that at the end I will be able to compare the real and imaginary parts of the ratio, but I got difficult algebraic expressions, so I figure out it's not the way. Can you assist please ?

Thanks !

Rather than the conjugate think about the sequence side of it instead - can you find a pair of expressions for the common ratio (and therefore equal to each other)?

 

[Yankel]

That's what I was trying to do, to find two expressions for the ratio. I need two equations somehow.

 

[skeeter]

That's what I was trying to do, to find two expressions for the ratio. I need two equations somehow.

$r= \dfrac{8+i}{m-3i} = \dfrac{n+17i}{8+i}$

$63+16i = (mn+51)+(17m-3n)i$

$mn=12$

$17m-3n=16$

one solution for the system is $m=2$, $n=6$

there is another possible, but I'm too lazy to check.