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

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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