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### 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.
### 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.
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