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, a_{1} = a = 12 + 8i, a_{2} = 11 + 6i

Common difference, d = a_{2} – a_{1}

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

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

∴ a_{n} = 12 + 8i + (n – 1) -1 – 2i

⇒ a_{n} = 12 + 8i – n – 2ni + 1 + 2i

⇒ a_{n} = 13 + 10i – n – 2ni

⇒ a_{n} = (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, 5^{th} 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, 13^{th} term is purely imaginary

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