0 sis....a.c,m[a+(m-1)d]=n[a+(n-1)d]....ma -na +m(m-1)d-n(n-1)d=0..(m-n)a+(m^2-m)d-(n^2-n)d=0...(m-n)a+(m^2-m-n^2-n)d=0..(m-n)a+[(m^2-n^2)-(m-n)]d=0...(m-n)a+[(m+n)(m-n)-1(m-n)]d=0..(m+n)[a+(m+n-1)d]=0..n henceforth (m+n)th term will be 0..hope u got this...

Understanding the Problem
In this problem, we need to find the term of a sequence based on a relationship between the m-th and n-th terms. Specifically, we know that the m times the m-th term is equal to the n times the n-th term.
Key Relationships
- Let the m-th term be denoted as T(m).
- Let the n-th term be denoted as T(n).
According to the problem, we can express this relationship mathematically as:
- m * T(m) = n * T(n)
This indicates that the value of the m-th term, when multiplied by m, equals the value of the n-th term, when multiplied by n.
Finding m+n-th Term
To find the (m+n)-th term, we can use the relationship established above.
- We can express T(m) and T(n) in terms of each other using the equation derived from the relationship.
- By rearranging, we can derive a general formula for T(m+n) based on the terms T(m) and T(n).
Assuming the sequence follows a specific pattern (e.g., arithmetic, geometric), we can substitute these values back into the term formula.
Conclusion
Once we establish the values of T(m) and T(n) through their relationship, we can easily compute T(m+n).
- The final answer will depend on the specific nature of the sequence involved (whether it is arithmetic, geometric, etc.).
For a more detailed exploration of sequences and their properties, consider visiting EduRev for comprehensive resources and examples.