if the teeth term of an ap is Q and the q3b speed then P + Q to term is

Answer: The (p+q)th(p + q)^{\text{th}} term is 00.

Step-by-step solution

Step 1: Set up equations using the general term formula

Let the first term of the arithmetic progression be aa and the common difference be dd. The formula for the nthn^{\text{th}} term is an=a+(n−1)da_n = a + (n - 1)d. Using this, the pthp^{\text{th}} term is a+(p−1)d=qa + (p - 1)d = q and the qthq^{\text{th}} term is a+(q−1)d=pa + (q - 1)d = p.

Step 2: Subtract equation (2) from equation (1) to find d

Subtracting equation (2) from equation (1) eliminates the first term aa. We get (p−q)d=q−p(p - q)d = q - p. Since p≠qp \neq q, dividing both sides by (p−q)(p - q) gives d=−1d = -1.

Step 3: Express the first term a in terms of p and q

Substitute d=−1d = -1 into equation (1). Expanding the expression gives a−p+1=qa - p + 1 = q. Rearranging terms to isolate aa yields a=p+q−1a = p + q - 1.

Step 4: Evaluate the (p + q)th term

The (p+q)th(p + q)^{\text{th}} term is given by ap+q=a+(p+q−1)da_{p+q} = a + (p + q - 1)d. Substituting a=p+q−1a = p + q - 1 and d=−1d = -1, the expression becomes (p+q−1)−(p+q−1)=0(p + q - 1) - (p + q - 1) = 0.

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