Question

if the Pth term of a. AP is q an the Qth term is p, then prove that its Nth term is (p+1-n) and hence prove that its(p+q)term is zero

Answer 1

According to question

Tp = q
a + (p-1)d = q -----------------(1)

Tq = p
a + (q-1)d = p -----------------(2)

On subtracting equation 1 from 2,we get

(q-1)d - (p-1)d = p-q
d(q-1-p+1) = (p-q)
d(q-p) = - 1(q-p)
d = - 1

Putting the value of d in equation 1, we get
a+(p-1) (-1) = q
a +(1-p) =q
a= p+q-1

Now,
Tn = a+ (n-1)d
= p+q-1+ (n-1) (-1)
= p+q-1 +(1-n)
= p+q-1+1-n
= p+q - n

Hence Proved



Hope it help you

Answer 2

ap = a + (p-1)d 
q = a + pd -d ...........1

aq = a + (q-1)d
p = a + qd - d............2

subtract 2 from 1 we get
q - p = a + pd -d - a - qd+ d
q - p = pd - qd
-1(p - q) = d(p - q)
d = -1

put value of d in 1 equation
q = a - p +1
a = p + q - 1

a(p+q) = a + (p+q - 1)d
put value of a = p + q - 1 and d= -1
           = p + q - 1 - p - q +1
           =  0

an = a + (n-1)d
     =p+ q - 1 - n + 1
     =p + q - n