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Question
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely imaginary?
Solution
12 + 8i, 11 + 6i, 10 + 4i...
This is an A.P.
Here, we have:
a = 12 + 8i
\[d = \left( 11 + 6i - 12 - 8i \right)\]
\[ = \left( - 1 - 2i \right)\]
\[\text { Let the imaginary term be } a_n = a + \left( n - 1 \right)d\]
\[ a_n = \left( 12 + 8i \right) + \left( n - 1 \right)\left( - 1 - 2i \right)\]
\[ = \left( 12 + 8i \right) + \left( - n + 1 - 2in + 2i \right)\]
\[ = 12 + 8i - n + 1 - 2in + 2i\]
\[ = \left( 13 - n \right) + \left( 8 - 2n + 2 \right)i\]
\[ = \left( 13 - n \right) + \left( 10 - 2n \right)i\]
\[ a_n\text { has to be imaginary } . \]
\[ \therefore \left( 13 - n \right) = 0\]
\[ \Rightarrow n = 13\]
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