Permutations and Combinations

Permutations and Combinations

Question

Which term of the sequence 12 + 8i, 11 + 6i, 10+41i, ....... is

(i) purely real   (ii) purely imaginary?

Answer

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

∴                           straight t subscript straight n space equals space straight a space plus space left parenthesis straight n space minus space 1 right parenthesis straight d 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

∴                  straight t subscript straight n space equals space straight a space plus space left parenthesis straight n space minus space 1 right parenthesis straight d 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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