In an A.P. the $p$ th term is $q$ and the $(p+q)^{\text {th }}$ term is 0 . Then the $q$ th term is (A) $-p$ (B $p$ (C) $p+q$ (D) $p-q$

Step 1: Define terms of an A.P.

Let $a$ be the first term and $d$ be the common difference of the Arithmetic Progression (A.P.). The $n$ -th term of an A.P. is given by the formula $a_n = a + (n-1)d$.

Step 2: Formulate equations from given conditions

According to the problem statement, the $p$ -th term is $q$, so we can write $a + (p-1)d = q$. Also, the $(p+q)$ -th term is $0$, which gives us the equation $a + (p+q-1)d = 0$. These are our two main equations.

Step 3: Solve for the common difference 'd'

To find the common difference $d$, we subtract equation (1) from equation (2). This eliminates $a$ and allows us to solve for $d$. After simplifying, we find that $d = -1$.

Step 4: Solve for the first term 'a'

Now that we have the value of $d$, we can substitute it back into equation (1) to find the first term $a$. Substituting $d = -1$ into $a + (p-1)d = q$, we get $a - p + 1 = q$, which simplifies to $a = p + q - 1$.

Step 5: Calculate the q-th term

Finally, we need to find the $q$ -th term, $a_q$. Using the formula $a_q = a + (q-1)d$ and substituting the values of $a$ and $d$ we found, we get $a_q = (p+q-1) + (q-1)(-1)$. This simplifies to $p+q-1-q+1$, which results in $p$.

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