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# If the ${{\text{p}}^{th}}$term of an A.P. is $q$ and ${{\text{q}}^{th}}$ term is $p$, prove that its ${{\text{n}}^{th}}$ term is $\left( {p + q - n} \right).$

Last updated date: 24th Mar 2023
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Hint- In an A.P ${{\text{n}}^{th}}$ Term is given as $a + \left( {n - 1} \right)d$ where $a$ is the first term and $d$ is the common difference of an A.P.

In the question above it is given that ${{\text{p}}^{th}}$term of an A.P. is $q$ and ${{\text{q}}^{th}}$ term is $p$ of an A.P.
For the given question ${{\text{n}}^{th}}$ Term of an A.P is asked, to find it we know in general form ${{\text{n}}^{th}}$ Term is given as $a + \left( {n - 1} \right)d$ where $a$ is the first term and $d$ is the common difference of an A.P.
So to solve this question first let us assume $a$ be the first term and $d$ is the common difference of the given Arithmetic progression.
So we can write ${{\text{p}}^{th}}$term and ${{\text{q}}^{th}}$ term of an A.P as
${{\text{p}}^{th}}{\text{ term }} = q \Rightarrow a + \left( {p - 1} \right)d = q{\text{ }}........\left( 1 \right)$
And similarly
${{\text{q}}^{th}}{\text{ term }} = p \Rightarrow a + \left( {q - 1} \right)d = p{\text{ }}........\left( 2 \right)$
From the above two equations we can find the value of $a$ and $d$ which we need to find the ${{\text{n}}^{th}}$ Term.
So, we will subtract equation (2) from (1), from here we will get $d$
$\left( {p - q} \right)d = \left( {q - p} \right) \Rightarrow d = - 1$
And now the value of $d$obtained above we will put in equation (1), from here we will get $a$ value
${\text{i}}{\text{.e }}a + \left( {p - 1} \right) \times \left( { - 1} \right) = q \Rightarrow a = \left( {p + q - 1} \right)$
So we need to find the ${{\text{n}}^{th}}$ Term
${{\text{n}}^{th}}$ Term $= a + \left( {n - 1} \right)d = \left( {p + q - 1} \right) + \left( {n - 1} \right) \times - 1 = \left( {p + q - n} \right)$
Hence Proved the ${{\text{n}}^{th}}$ term is $\left( {p + q - n} \right).$

Note- Whenever this type of question appears it is important to note down given details as in this question it is given ${{\text{p}}^{th}}$term of an A.P. is $q$ and ${{\text{q}}^{th}}$ term is $p$. In Arithmetic Progression the difference between the two successive terms is same and we call it common difference $d$.In an A.P ${{\text{n}}^{th}}$ Term is given as $a + \left( {n - 1} \right)d$ where $a$ is the first term and $d$ is the common difference of an A.P. Approach this type of question with intent to find the value of $a$and $d$.