Question

Which term of the sequence $\left( { - 8 + 18i} \right),\left( { - 6 + 15i} \right),( - 4 + 12i)......$ is purely imaginary?
$\eqalign{
  & 1)5th \cr
  & 2)7th \cr
  & 3)8th \cr
  & 4)6th \cr} $

Answer 1

Hint: The given question contains terms in a sequence which has a pattern. We need to figure out the pattern in which the terms are placed. We are provided with only three terms and these are not purely imaginary as they contain a real part. So, to find out a term which is purely imaginary, we need to find out the next terms in the sequence. As we go on, we can find out the answer.

Complete step-by-step answer:
The given sequence is $\left( { - 8 + 18i} \right),\left( { - 6 + 15i} \right),( - 4 + 12i)......$
We observe that there is a constant increase in both the real and the imaginary part of the terms.
Therefore, to find the common difference between the terms, we just need to subtract the first one from the second.
Common difference, $d = \left( { - 6 + 15i} \right) - ( - 8 + 18i)$
The real and the imaginary parts are added and subtracted separately.
$\eqalign{
  & \Rightarrow \left( { - 6 + 8} \right) + \left( {15i - 18i} \right) \cr
  & = 2 - 3i \cr} $
Now, to find the next consecutive terms, we can go on adding the common difference to the previous terms
The fourth term will be,
$\eqalign{
  & = - 4 + 12i + 2 - 3i \cr
  & = - 2 + 9i \cr} $
This has both a real and an imaginary part. So, we move on to finding the next term
The fifth term is
$\eqalign{
  & = - 2 + 9i + 2 - 3i \cr
  & = 6i \cr} $
Since there is no real part, the fifth term is purely imaginary.
Hence, option (1) is the correct answer.
So, the correct answer is “Option 1”.

Note: The real part is just a real number and the imaginary part contains a real number with $i$. These terms have to be added separately. Remember that a real part cannot be added or subtracted from an imaginary part and vice versa. So, be careful while carrying out those operations.