Question

Which term of the sequence 12 + 8i. 11 + 6i, 10 + 4i, ..... is (a) purely real (b) purely imaginary?​

Answer 1

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.

Answer 2

Answer:  

The answer is given in the attachment