A recreational series problem

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In summary, the conversation discusses the problem of finding the input impedance of a series of cascaded resistors and how to approach it. The main question is how to get rid of the product and summation terms in order to find an expression for Zn+1, R1, R2, and n. The suggestion is to use the limit as Zn+1 approaches Zn to find the final expression.
  • #1
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Dear all,

The problem of finding the input impedance of a series of cascaded resisters can be stated as

[tex]Z_{n+1}=R_1+\frac{R_2 Z_n}{R_2+Z_n} [/tex] where [tex]Z_1=R_1+R_2[/tex]. What is [tex]\lim_{n\to\infty}Z_n[/tex]?

My attempt is to re-write the recurrance relation as
[tex](R_2+Z_n)Z_{n+1}-(R_1+R_2)Z_n-R_1R_2=0[/tex]

which is
[tex]R_2 Z_{n+1} + Z_{n}Z_{n+1} - (R_1+R_2)Z_n - R_1 R_2 = 0[/tex]

[tex]R_2 Z_{n} + Z_{n-1}Z_{n} - (R_1+R_2)Z_{n-1} - R_1 R_2 = 0[/tex]
...
[tex]R_2 Z_2 + Z_1 Z_2 - (R_1+R_2)Z_1 - R_1 R_2 = 0[/tex]

Summing them up gives
[tex]R_2(Z_{n+1}-Z_1)+Z_{n+1}Z_n + ... + Z_2 Z_1 - R_1(Z_n +...+Z_1) - nR_1R_2=0[/tex].

I am not sure on how to get rid of the product terms and summation terms to get an expression of only [tex]Z_{n+1}, R_1, R_2 [/tex] and [tex]n[/tex]. Any suggestion on possible attack?

Thank you.

elgen
 
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  • #2
hi elgen! :smile:

the question only asks for the limit

you can find the limit (if it exists) by putting Zn+1 = Zn :wink:
 
  • #3
That sounds cool. Thx for the pointer!
 

1. What is a recreational series problem?

A recreational series problem is a type of mathematical puzzle or game that is meant to be solved for enjoyment rather than practical application. These problems often involve logic, pattern recognition, and creative thinking.

2. How difficult are recreational series problems?

The difficulty level of recreational series problems varies greatly. Some may be relatively simple and only require basic math skills, while others may be extremely challenging and require advanced mathematical concepts. It ultimately depends on the specific problem and the individual's skill level.

3. Can recreational series problems help improve cognitive skills?

Yes, solving recreational series problems can help improve cognitive skills such as critical thinking, problem-solving, and pattern recognition. These skills can be useful in many areas of life, not just in mathematics.

4. Are there any resources or tools available for solving recreational series problems?

Yes, there are various resources and tools available for solving recreational series problems, such as books, online forums, and problem-solving techniques. These can help individuals approach problems in a more structured and efficient way.

5. Are there any benefits to solving recreational series problems?

Aside from improving cognitive skills, solving recreational series problems can also provide a sense of satisfaction and accomplishment. It can also be a fun and challenging way to pass the time and exercise the brain.

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