Question

Which term of the sequence$\left(-8+18i\right),\left(-6+15i\right),\left(-4+12i\right),.....$ is purely imaginary

• A. $5^{th}$term
• B. $7^{th}$term
• C. $8^{th}$term
• D. $6^{th}$term

Answer 1

The correct option is A

$5^{th}$term

Explanation of correct option:

Step 1: Note the given data

The series is $\left(-8+18i\right),\left(-6+15i\right),\left(-4+12i\right),.....$

$a_1=\left(-8+18i\right)$; $d:\left(-6+15i\right)-\left(-8+18i\right)=2-3i$

Since the difference is constant, the series is in arithmetic progression.

Step 2: Find the term that is purely imaginary

The $n^{th}$ term of an AP is $t_n=a+\left(n-1\right)d$, where $a$ is first term, $n$ is the number of terms and $d$ is common term.

Let $n^{th}$ term of the AP be purely imaginary.

The $n^{th}$ term of the AP is $-8+18i+\left(n-1\right)\left(2-3i\right)$

$=-8+\left(n-1\right)2+18i-\left(n-1\right)3i$

So,$-8+\left(n-1\right)2=0$

$\Rightarrow$$-8+2n-2=0$

$\Rightarrow$ $-10+2n=0$

$\Rightarrow$ $2n=10$

$\Rightarrow$ $n=5$

Thus $5^{th}$term of the AP is purely imaginary.

Hence the correct option is A.