Q. 435.0( 4 Votes )

If the pth term of an A.P. is x and qth term is y, show that the sum of (p + q) terms is

Answer :

Given: ap = x and aq = y


We know that,


an = a + (n – 1)d


ap = a + (p – 1)d


x = a + (p – 1)d …(i)


Now,


aq = a + (q – 1)d


y = a + (q – 1)d …(ii)


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


x – (p – 1)d = y – (q – 1)d


x – y = (p – 1)d – (q – 1)d


x – y = d [p – 1 – q + 1]


x – y = d[ p – q]


…(iii)


Adding, Eq (i) and (ii), we get


x + y = 2a + (p – 1) + (q – 1)d


x + y = 2a + d[p + q – 1 – 1]


x + y = 2a + d (p + q – 1) –d


x + y + d = 2a + (p + q – 1)d …(iv)


We know that,




[using (iv)]


[using (iii)]


Hence Proved


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