MHB Understanding Complex Geometric Sequences: A Revision Question

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The discussion focuses on solving a revision question related to complex geometric sequences. The sequence is defined as \( u_n = 3(1+i)^{n-1} \), leading to the product sequence \( v_n = u_n u_{n+k} = 9(1+i)^{2n+k-2} \). It is established that \( v_n \) is also a geometric sequence, as shown by the consistent ratio \( \frac{v_n}{v_{n-1}} = (1+i)^2 \). The conversation emphasizes the importance of understanding the formulas for the sum of geometric sequences for further questions. Additional resources, such as Wikipedia, are suggested for further clarification.
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need a hand with a revision question, I don't quite understand how to go about solving it question is attached below
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Hi, and welcome to the forum!

According to the definition of $u_n$,
\[
u_4=(1+i)u_3=(1+i)^2u_2=(1+i)^3u_1=3(1+i)^3=-6+6i.
\]
In general, $u_n=3(1+i)^{n-1}$. From there
\[
v_n=u_nu_{n+k}=3(1+i)^{n-1}\cdot3(1+i)^{n+k-1}=9(1+i)^{2n+k-2}.
\]
In particular,
\[
v_{n+1}=9(1+i)^{2n+k}=(1+i)^2v_n
\]
which means that $v_n$ is also a geometric sequence.

Another way to show this is to note that $\dfrac{u_n}{u_{n-1}}=1+i$ for all $i$. Therefore
\[
\dfrac{v_n}{v_{n-1}}=\dfrac{u_nu_{n+k}}{u_{n-1}u_{n+k-1}}=
\dfrac{u_n}{u_{n-1}}\cdot\dfrac{u_{n+k}}{u_{n+k-1}}=(1+i)(1+i)=(1+i)^2.
\]
The ratio does not depend on $n$; therefore, $v_n$ is a geometric sequence.

To answer other questions, you need to know formulas for the sum of geometric sequences and other information, which you can find in Wikipedia.

Please also read the http://mathhelpboards.com/rules/, especially rules 8 and 11.
 
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