Question

If m times the $ {{m}^{th}} $ term of an A.P is equal to the n times the $ {{n}^{th}} $ term, then show that the $ {{\left( m+n \right)}^{th}} $ term of the A.P is zero.

Answer 1

Hint: We start solving the problem by assuming variables for first term and the common difference of the given A.P. We then recall the fact that the $ {{r}^{th}} $ term of the A.P is defined as $ {{T}_{r}}=a+\left( r-1 \right)d $ . We then use the condition given in the problem: “m times the $ {{m}^{th}} $ term of an A.P is equal to the n times the $ {{n}^{th}} $ term” using the $ {{r}^{th}} $ term. We then make the necessary calculations to prove the required result.

Complete step by step answer:
According to the problem, we are given that m times the $ {{m}^{th}} $ term of an A.P is equal to the n times the $ {{n}^{th}} $ term. We need to show that the $ {{\left( m+n \right)}^{th}} $ term of the A.P is zero.
Let us assume the first term of the given A.P as a and the common difference as d.
We know that the $ {{r}^{th}} $ term of the A.P is defined as $ {{T}_{r}}=a+\left( r-1 \right)d $ .
According to the problem, we are given $ m{{T}_{m}}=n{{T}_{n}} $ .
 $ \Rightarrow m\left( a+\left( m-1 \right)d \right)=n\left( a+\left( n-1 \right)d \right) $ .
 $ \Rightarrow ma+\left( {{m}^{2}}-m \right)d=na+\left( {{n}^{2}}-n \right)d $ .
 $ \Rightarrow ma-na+\left( {{m}^{2}}-m \right)d-\left( {{n}^{2}}-n \right)d=0 $ .
 $ \Rightarrow \left( m-n \right)a+\left( {{m}^{2}}-m-{{n}^{2}}+n \right)d=0 $ .
 $ \Rightarrow \left( m-n \right)a+\left( \left( m-n \right)\left( m+n \right)-1\left( m-n \right) \right)d=0 $ .
 $ \Rightarrow \left( m-n \right)\left( a+\left( m+n-1 \right)d \right)=0 $ .
 $ \Rightarrow a+\left( m+n-1 \right)d=0 $ .
 $ \Rightarrow {{T}_{m+n}}=0 $ .
So, we have found that the value of the $ {{\left( m+n \right)}^{th}} $ term of the A.P is zero.
 $ \therefore $ We have proved that the value of the $ {{\left( m+n \right)}^{th}} $ term of the A.P is zero.

Note:
 Whenever we get this type of problem, we try to assign the variable to the unknowns present in the problem to avoid confusion in the problem. We should not make calculation mistakes while solving this problem. Here $ \left( m-n \right) $ cannot be equal to zero as we assumed that the $ {{m}^{th}} $ term and $ {{n}^{th}} $ term are different before solving this problem. Similarly, we can expect problems to find the relation between first term and common difference using the obtained result.