Q. 165.0( 4 Votes )

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

Answer :

Given A.P is 12 + 8i, 11 + 6i, 10 + 4i, …


Here, a1 = a = 12 + 8i, a2 = 11 + 6i


Common difference, d = a2 – a1


= 11 + 6i – (12 + 8i) = 11 – 12 + 6i – 8i = -1 – 2i


We know, an = a + (n – 1)d where a is first term or a1 and d is common difference and n is any natural number


an = 12 + 8i + (n – 1) -1 – 2i


an = 12 + 8i – n – 2ni + 1 + 2i


an = 13 + 10i – n – 2ni


an = (13 – n) + (10 – 2n)i


To find purely real term of this A.P., imaginary part have to be zero


10 – 2n = 0


2n = 10



n = 5


Hence, 5th term is purely real


To find purely imaginary term of this A.P., real part have to be zero


13 – n = 0


n = 13


Hence, 13th term is purely imaginary

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