Sure, let's go through the steps to solve the problem clearly.
### Part (a)
Given:
1. The [tex]\( p \)[/tex]-th term of an arithmetic sequence (A.S) is [tex]\( q \)[/tex].
2. The [tex]\( q \)[/tex]-th term is [tex]\( p \)[/tex].
We need to show that the [tex]\( m \)[/tex]-th term is [tex]\( p + q - m \)[/tex].
Let's denote the first term of the arithmetic sequence by [tex]\( a \)[/tex] and the common difference by [tex]\( d \)[/tex].
The general formula for the [tex]\( n \)[/tex]-th term of an A.S. is:
[tex]\[ T_n = a + (n-1)d \][/tex]
Given:
1. [tex]\( T_p = q \)[/tex]
2. [tex]\( T_q = p \)[/tex]
Using the formula for the [tex]\( p \)[/tex]-th term:
[tex]\[ a + (p-1)d = q \tag{1} \][/tex]
Using the formula for the [tex]\( q \)[/tex]-th term:
[tex]\[ a + (q-1)d = p \tag{2} \][/tex]
Now, subtract equation (1) from (2):
[tex]\[ (a + (q-1)d) - (a + (p-1)d) = p - q \][/tex]
[tex]\[ (q-1)d - (p-1)d = p - q \][/tex]
[tex]\[ (q-p)d = p - q \][/tex]
[tex]\[ d = \frac{p-q}{q-p} \][/tex]
[tex]\[ d = -1 \][/tex]
So, we find that [tex]\( d = -1 \)[/tex].
Now substitute [tex]\( d = -1 \)[/tex] back into equation (1):
[tex]\[ a + (p-1)(-1) = q \][/tex]
[tex]\[ a - (p-1) = q \][/tex]
[tex]\[ a = q + p - 1 \][/tex]
We need to find the [tex]\( m \)[/tex]-th term:
[tex]\[ T_m = a + (m-1)d \][/tex]
Substitute the values of [tex]\( a \)[/tex] and [tex]\( d \)[/tex]:
[tex]\[ T_m = (q + p - 1) + (m-1)(-1) \][/tex]
[tex]\[ T_m = q + p - 1 - m + 1 \][/tex]
[tex]\[ T_m = p + q - m \][/tex]
Therefore, we have shown that the [tex]\( m \)[/tex]-th term is [tex]\( p + q - m \)[/tex].
### Part (b)
Let's denote the [tex]\( m \)[/tex]-th term by [tex]\( T_m \)[/tex] and the [tex]\( n \)[/tex]-th term by [tex]\( T_n \)[/tex].
Given:
1. [tex]\( m \times T_m = n \times T_n \)[/tex]
Using the general formula again:
[tex]\[ T_m = a + (m-1)d \][/tex]
[tex]\[ T_n = a + (n-1)d \][/tex]
Substitute these into the given condition:
[tex]\[ m(a + (m-1)d) = n(a + (n-1)d) \][/tex]
Expand:
[tex]\[ ma + m(m-1)d = na + n(n-1)d \][/tex]
Rearrange to group terms involving [tex]\( a \)[/tex] and [tex]\( d \)[/tex]:
[tex]\[ ma - na = n(n-1)d - m(m-1)d \][/tex]
[tex]\[ a(m-n) = d[n(n-1) - m(m-1)] \][/tex]
Notice that:
[tex]\[ n(n-1) - m(m-1) = n^2 - n - (m^2 - m) \][/tex]
[tex]\[ = n^2 - n - m^2 + m \][/tex]
Thus, the equation becomes:
[tex]\[ a(m-n) = d(n^2 - n - m^2 + m) \][/tex]
This shows the relationship required in part (b).
