Which term of the sequence 12+8i, 6i, 10+4i, ... is (a) real (b) … - Vidyadip

Question

Which term of the sequence $12+8\text{i},\ 6\text{i},\ 10+4\text{i},\ ...$ is $(a)$ real $(b)$ purely imaginary$?$

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Answer

The given sequence is $12+8\text{i},\ 6\text{i},\ 10+4\text{i},\ ...$
Here, $\text{a}=12+8\text{i}$
$\text{d}=-1-2\text{i}$
Then, $\text{a}_\text{n}=\text{a}(\text{n}-1)\text{d}$
$=12+8\text{i}+(\text{n}-1)(-1-2\text{i})$
$=(13-\text{n})+\text{i}(10-2\text{i})$
Let $n^{th}$ term be purely real the $(10-2\text{n})=0$ or $\text{n}=5$
So, $5$th term is purely real.
Let $n^{th}$ term be purely im againarym than, $13-\text{n}=0$
$\therefore\text{n}=13$
So, $13$th term is purely inaginary.

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