Let a be the first term and d be the common difference of the AP. Using the formula for the n^(th) term, a_n = a + (n-1)d, we formulate two linear equations corresponding to the p^(th) and q^(th) terms.

Let a be the first term and d be the common difference of the AP. Using the formula for the n^(th) term, a_n = a + (n-1)d, we formulate two linear equations corresponding to the p^(th) and q^(th) terms.

if the (p)th term of an ap is (q) and the (q)th term is (p) then its (p+q)th term is

Answer: 0

Step-by-step solution

Step 1: State formula and write given equations

Let $a$ be the first term and $d$ be the common difference of the AP. Using the formula for the $n^{\text{th}}$ term, $a_n = a + (n-1)d$ , we formulate two linear equations corresponding to the $p^{\text{th}}$ and $q^{\text{th}}$ terms.

Step 2: Solve for common difference d

Subtract equation (2) from equation (1) to eliminate the first term $a$ . This gives $(p-q)d = -(p-q)$ . Since $p \neq q$ , dividing both sides by $(p-q)$ yields $d = -1$ .

Step 3: Find the first term a

Substitute $d = -1$ into equation (1). Solving for $a$ gives $a = p + q - 1$ .

Step 4: Calculate the (p + q)-th term

Now we compute the $(p+q)^{\text{th}}$ term using $a_{p+q} = a + (p+q-1)d$ . Substituting $a = p+q-1$ and $d = -1$ , the terms cancel out to yield $0$ .

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