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Choose whether the given statement is true or false.

If m times the $m^{th}$ 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.

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True

False

answer is A.

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Detailed Solution

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

{(m + n)}^{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^{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+(m−1)d) = n(a+(n−1)d)
⇒ ma+(

m^{2}

−m)d = na+(

n^{2}

−n)d
⇒ ma−na+(

m^{2}

−m)d−(

n^{2}

−n)d = 0
⇒ (m−n)a+(

m^{2}

−m−

n^{2}

+n)d = 0
⇒ (m−n)a+((m−n)(m+n)−1(m−n))d = 0
⇒ (m−n)(a+(m+n−1)d) = 0
⇒ a+(m+n−1)d = 0
⇒

T_{m + n} = 0

So, we have found that the value of the

{(m + n)}^{th}

term of the A.P is zero.
∴ We have proved that the value of the

{(m + n)}^{th}

term of the A.P is zero
So, the given statement is true.

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