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Permutations And Combinations
Which term of the sequence 12 + 8i, 11 + 6i, 10+41i, ....... is
(i) purely real (ii) purely imaginary?
The given sequence is :
12 + 8i, 11 + 6i, 10+4i, .......
Here, a = 12 + 8i
d = (11 + 6i)  (12 + 8i) = 1  2i
(i) Let nth term be purely real
∴ is purely real
or 12 + 8i + (n  1) (1  2i) is purely real
or 12 + 8i  (n  1)  2 (n  1) is purely real
or (13  n) + (10  2n) i is purely real
∴ 10  2n = 0 or n = 5
∴ 5th term is purely real.
(ii) Let nth term be purely imaginary
∴ is purely imaginary
or (13  n) + (10  2n) is purely imaginary
or 13  n = 0
or n = 13
∴ 13th term is purely imaginary.
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