Which term of the A.P. (16-6i,)(15-4i), (14-2 i), ... is a :
(a) pure real number ?
(b) pure imaginary number ?

Explanation

To solve the problem, we will analyze the given arithmetic progression (A.P.) and determine the terms that are purely real and purely imaginary.

### Given A.P.:
The terms of the A.P. are:
1. \( a_1 = 16 - 6i \)
2. \( a_2 = 15 - 4i \)
3. \( a_3 = 14 - 2i \)
4. ...

### Step 1: Identify the first term and common difference
- The first term \( a = 16 - 6i \).
- To find the common difference \( d \), we calculate:
\[ d = a_2 - a_1 = (15 - 4i) - (16 - 6i) = 15 - 4i - 16 + 6i = -1 + 2i \]

### Step 2: Write the nth term formula
The nth term \( a_n \) of an A.P. is given by:
\[ a_n = a + (n - 1)d \]
Substituting the values of \( a \) and \( d \):
\[ a_n = (16 - 6i) + (n - 1)(-1 + 2i) \]
Expanding this:
\[ a_n = 16 - 6i + (n - 1)(-1) + (n - 1)(2i) \]
\[ = 16 - 6i - (n - 1) + (n - 1)(2i) \]
\[ = (16 - (n - 1)) + (-6 + 2(n - 1))i \]
\[ = (17 - n) + (2n - 8)i \]

### Step 3: Find when the term is purely real
For \( a_n \) to be a purely real number, the imaginary part must be zero:
\[ 2n - 8 = 0 \]
Solving for \( n \):
\[ 2n = 8 \implies n = 4 \]
Thus, the 4th term is purely real.

### Step 4: Find when the term is purely imaginary
For \( a_n \) to be a purely imaginary number, the real part must be zero:
\[ 17 - n = 0 \]
Solving for \( n \):
\[ n = 17 \]
Thus, the 17th term is purely imaginary.

### Final Answers:
(a) The term that is a pure real number is the **4th term**.
(b) The term that is a pure imaginary number is the **17th term**.