To solve the given problem, let's start by understanding the properties of an Arithmetic Progression (A.P.).
Given:
- The [tex]\( p \)[/tex]-th term of the A.P. is [tex]\( q \)[/tex]: [tex]\( a + (p - 1)d = q \)[/tex]
- The [tex]\( q \)[/tex]-th term of the A.P. is [tex]\( p \)[/tex]: [tex]\( a + (q - 1)d = p \)[/tex]
where:
- [tex]\( a \)[/tex] is the first term of the A.P.
- [tex]\( d \)[/tex] is the common difference.
We need to find the [tex]\((p + q)\)[/tex]-th term of this A.P.
First, let's set up the initial equations:
1. [tex]\( a + (p - 1)d = q \)[/tex]
2. [tex]\( a + (q - 1)d = p \)[/tex]
We will subtract the second equation from the first to eliminate [tex]\( a \)[/tex]:
[tex]\[ (a + (p - 1)d) - (a + (q - 1)d) = q - p \][/tex]
[tex]\[ a + (p - 1)d - a - (q - 1)d = q - p \][/tex]
[tex]\[ (p - 1)d - (q - 1)d = q - p \][/tex]
[tex]\[ (p - 1 - q + 1)d = q - p \][/tex]
[tex]\[ (p - q)d = q - p \][/tex]
Since [tex]\( p - q \)[/tex] is not zero, we can divide both sides by [tex]\( p - q \)[/tex]:
[tex]\[ d = \frac{q - p}{p - q} \][/tex]
[tex]\[ d = -1 \][/tex]
Now that we have the common difference [tex]\( d = -1 \)[/tex], substitute this back into either of the original equations to find [tex]\( a \)[/tex]. Using the first equation:
[tex]\[ a + (p - 1)(-1) = q \][/tex]
[tex]\[ a - p + 1 = q \][/tex]
[tex]\[ a = q + p - 1 \][/tex]
Now, we need to find the [tex]\((p + q)\)[/tex]-th term of the A.P.:
[tex]\[ \text{(p + q)-th term} = a + (p + q - 1)d \][/tex]
Substitute [tex]\( a \)[/tex] and [tex]\( d \)[/tex]:
[tex]\[ \text{(p + q)-th term} = (q + p - 1) + (p + q - 1)(-1) \][/tex]
[tex]\[ = (q + p - 1) + (-p - q + 1) \][/tex]
[tex]\[ = q + p - 1 - p - q + 1 \][/tex]
[tex]\[ = 0 \][/tex]
Therefore, the [tex]\((p + q)\)[/tex]-th term is [tex]\( \boxed{0} \)[/tex].
So the correct answer is:
c. 0
