Given ap  = Q        aQ =  pWe know that an = a +(n-1)d         Then    ap= a +(p-1)d                        Q=  a +(p-1)d      ...................(1)               aQ =a +(Q-1)d                p= a+ (Q-1)d      .....................(2)      SUBTRACTING (2)FROM (1 ) WE GETQ-p = a -a +(p-1)d  -(Q-1)d  Q-p       =  (pd -d) -(Qd -d)    Q-p   = pd -Qd -d+d     Q-p =  (p-Q)d(Q -p) = -(Q-p)d-d =(Q-p) / (Q-p)=1d = -1   .........................(A)Nowap+Q = a +(p+Q -1)d           = a + (pd+Qd- d)          = a+ (pd -d) +Qd       =  a +(p -1)d + Qd    =    ap  +Qd   (FROM 1)              = Q + Qd         = Q +Q(-1)    (FROM A)    =Q-Q   =0

Proof:

Let's consider an arithmetic progression (AP) with the first term as 'a' and the common difference as 'd'.

The qth term of the AP can be written as:
P = a + (q-1)d ...(1)

Similarly, the pth term of the AP can be written as:
q = a + (p-1)d ...(2)

Solving for 'a' in equation (1):
a = P - (q-1)d ...(3)

Substituting the value of 'a' from equation (3) into equation (2), we get:
q = (P - (q-1)d) + (p-1)d

Simplifying the equation, we have:
q = P + (p-1)d - (q-1)d
q = P + pd - d - qd + d
q = P + pd - qd
q = P + d(p-q)

Rearranging the equation, we obtain:
d(p-q) = q - P

Now, let's find the (p+q)th term of the AP:

The (p+q)th term of the AP can be written as:
T = a + ((p+q)-1)d

Substituting the value of 'a' from equation (3), we get:
T = (P - (q-1)d) + ((p+q)-1)d
T = P + pd - d - qd + pd + qd - d
T = P + 2pd - 2d

Simplifying further, we have:
T = P - 2d + 2pd
T = P(1 + 2p) - 2d(1 - p)

Using the equation (p-q)d = q - P:

We can rewrite the equation as:
d(p-q) = q - P

Substituting the value of d from the equation into the (p+q)th term equation, we have:
T = P(1 + 2p) - 2(q - P)(1 - p)

Expanding and simplifying the equation, we get:
T = P + 2pP - 2q + 2Pp + 2q - 2Pp
T = P + 2pP - 2q + 2q - 2Pp

The terms 2Pp and -2Pp cancel out, leaving us with:
T = P - 2q + 2pP + 2q - 2q
T = P

Therefore, the (p+q)th term of the AP is equal to P, which implies that the (p-q)th term is zero.

Hence, the (p-q)th term of the given AP is zero.