Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, … is (a) purel

1.	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, a_n = a + (n – 1)d Where a is first term or a₁ 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 = \(\frac{10}{2}\) ⇒ 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.

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,

a_n = a + (n – 1)d

Where a is first term or a₁ 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 = \(\frac{10}{2}\)

⇒ 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.