Choose whether the given statement is true or false.If m times the mth term of an A.P is equal to the n times the n th term, then is it possible to show that the (m+n)th The term of the A.P is zero.

# Choose whether the given statement is true or false.If m times the ${m}^{\mathit{th}}$ term of an A.P is equal to the n times the  term, then is it possible to show that the $\left({m+n\right)}^{\mathit{th}}$ The term of the A.P is zero.

1. A
True
2. B
False

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### Solution:

According to the problem, we are given that m times the ${m}^{\mathit{th}}$ term of an A.P is equal to the n times the ${n}^{\mathit{th}}$ term. We need to show that the ${\left(m+n\right)}^{\mathit{th}}$ the 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 ${r}^{\mathit{th}}$ term of the A.P is defined as Tr= a+(r-1)d .
According to the problem, we are given mTm=nTn.
m(a+(m1)d) = n(a+(n1)d)
ma+(${m}^{2}$−m)d = na+(${n}^{2}$−n)d
mana+(${m}^{2}$−m)d−(${n}^{2}$−n)d = 0
(mn)a+(${m}^{2}$−m−${n}^{2}$+n)d = 0
(mn)a+((mn)(m+n)1(mn))d = 0
(mn)(a+(m+n1)d) = 0
a+(m+n1)d = 0

So, we have found that the value of the ${\left(m+n\right)}^{\mathit{th}}$ term of the A.P is zero.
We have proved that the value of the ${\left(m+n\right)}^{\mathit{th}}$ term of the A.P is zero
So, the given statement is true.

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