If [tex]\( m \)[/tex] times the [tex]\( m^{\text{th}} \)[/tex] term of an A.P. is equal to [tex]\( n \)[/tex] times its [tex]\( n^{\text{th}} \)[/tex] term, show that the [tex]\( (m \cdot n)^{\text{th}} \)[/tex] term of the A.P. is zero.



Answer :

Certainly! Let's tackle this problem step-by-step.

### Given:
- The [tex]\( m \)[/tex]-th term of an Arithmetic Progression (A.P.) is [tex]\( m \times \)[/tex] term [tex]\( T_m \)[/tex].
- The [tex]\( n \)[/tex]-th term of the A.P. is [tex]\( n \times \)[/tex] term [tex]\( T_n \)[/tex].
- It is given that [tex]\( m \times T_m = n \times T_n \)[/tex].

### Step-by-Step Solution:

1. Express the Terms in A.P.:

Let's denote the first term of the A.P. as [tex]\( a \)[/tex] and the common difference as [tex]\( d \)[/tex].

The general term [tex]\( T_k \)[/tex] of an A.P. can be written as:
[tex]\[ T_k = a + (k-1)d \][/tex]

2. Write [tex]\( T_m \)[/tex] and [tex]\( T_n \)[/tex]:

For the [tex]\( m \)[/tex]-th term:
[tex]\[ T_m = a + (m-1)d \][/tex]

For the [tex]\( n \)[/tex]-th term:
[tex]\[ T_n = a + (n-1)d \][/tex]

3. Express the Given Condition:

According to the problem, [tex]\( m \)[/tex] times the [tex]\( m \)[/tex]-th term equals [tex]\( n \)[/tex] times the [tex]\( n \)[/tex]-th term. Therefore:
[tex]\[ m \times T_m = n \times T_n \][/tex]

Substituting the terms:
[tex]\[ m \times (a + (m-1)d) = n \times (a + (n-1)d) \][/tex]

4. Expand and Simplify:

Distribute [tex]\( m \)[/tex] and [tex]\( n \)[/tex] respectively:
[tex]\[ ma + m(m-1)d = na + n(n-1)d \][/tex]

Simplify the equation:
[tex]\[ ma + m^2d - md = na + n^2d - nd \][/tex]

Rearrange the terms on each side:
[tex]\[ ma + m^2d - md = na + n^2d - nd \][/tex]

5. Combine Like Terms:

Moving all terms involving [tex]\( a \)[/tex] and [tex]\( d \)[/tex] to one side:
[tex]\[ ma - na + m^2d - md - n^2d + nd = 0 \][/tex]

Factor out [tex]\( a \)[/tex] and [tex]\( d \)[/tex]:
[tex]\[ (m - n)a + (m^2 - m)d - (n^2 - n)d = 0 \][/tex]

Combine like terms involving [tex]\( d \)[/tex]:
[tex]\[ (m - n)a + (m^2 - m - n^2 + n)d = 0 \][/tex]

Notice that:
[tex]\[ m^2 - m - n^2 + n = (m^2 - n^2) - (m - n) \][/tex]

Factor further:
[tex]\[ (m - n)a + (m + n)(m - n)d - (m - n)d = 0 \][/tex]

6. Simplify Further:

Factoring out [tex]\( (m - n) \)[/tex]:
[tex]\[ (m - n)\left[a + (m+n)d - d \right] = 0 \][/tex]

Since [tex]\( m \neq n \)[/tex]:
[tex]\[ a + (m+n)d - d = 0 \][/tex]

Simplify:
[tex]\[ a + md + nd - d = 0 \][/tex]

Simplify further:
[tex]\[ a + (m+n-1)d = 0 \][/tex]

7. Find the [tex]\( mn \)[/tex]-th Term:

Now, we need to show that the [tex]\( (mn) \)[/tex]-th term of the A.P. is zero. The [tex]\( (mn) \)[/tex]-th term [tex]\( T_{mn} \)[/tex] is:
[tex]\[ T_{mn} = a + (mn-1)d \][/tex]

8. Substitute and Simplify:

From the previous condition, we have [tex]\( a = - (m+n-1)d \)[/tex]. Substitute this in [tex]\( T_{mn} \)[/tex]:
[tex]\[ T_{mn} = - (m+n-1)d + (mn-1)d \][/tex]

Combine the terms:
[tex]\[ T_{mn} = (mn - 1 - m - n + 1)d \][/tex]

Simplify the expression:
[tex]\[ T_{mn} = (mn - m - n)d \][/tex]

Recall from our condition, [tex]\( a + (m+n-1)d = 0 \)[/tex], so:
[tex]\[ a = -(m+n-1)d \][/tex]

9. Observe Zero Result:

Substitute [tex]\( a \)[/tex]:
[tex]\[ a + (mn-1)d = -(m+n-1)d + (mn-1)d \][/tex]

Simplify:
[tex]\[ T_{mn} = (mn - 1 - m - n + 1)d = (mn - m - n)d = 0 \][/tex]

Therefore, we have shown that the [tex]\( (mn) \)[/tex]-th term of the Arithmetic Progression is zero:
[tex]\[ T_{mn} = 0 \][/tex]

This completes the proof.