MHB Geometric Series with Complex Numbers

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The discussion revolves around finding the real numbers n and m in a geometric series represented by the elements m-3i, 8+i, and n+17i. The user initially attempted to use the conjugate method to compare the ratios but encountered complex algebraic expressions. A suggestion was made to derive two expressions for the common ratio, leading to the equations r = (8+i)/(m-3i) and r = (n+17i)/(8+i). Solving the resulting system of equations, it was found that one solution is m=2 and n=6, with a note that another solution may exist. The conversation highlights the importance of finding equivalent expressions for the common ratio in geometric series involving complex numbers.
Yankel
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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 !
 
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Yankel said:
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)?
 
That's what I was trying to do, to find two expressions for the ratio. I need two equations somehow.
 
Yankel said:
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.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
View full post »

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