Answer :

Given: ap = q and aq = p


To Prove: an = (p + q – n)


Proof:


We know that,


an = a + (n – 1)d


Putting n = p


ap = a + (p – 1)d


q = a + (p – 1)d


q – (p – 1)d = a …(i)


Similarly,


aq = a + (q – 1)d


p = a + (q – 1)d


p – (q – 1)d = a …(ii)


From eq. (i) and (ii), we get


q – (p – 1)d = p – (q – 1)d


q – pd + d = p – qd + d


qd – pd = p – q


d(q – p) = -(q – p)


d = -1


Putting the value of d in eq. (i), we get


q – (p – 1)(-1) = a


q + p – 1 = a


Now, we have to find nth term


We know that,


an = a + (n – 1)d


Putting a = (q + p – 1) and d = -1


an = (p + q – 1) + (n – 1)(-1)


an = p + q – 1 – n + 1


an = p + q – n


Hence Proved


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