Question

If the sum of P terms of an A.P. is q and the sum of q terms is p, then the sum of p + q terms will be

Options

• 0

• p − q

•  p + q

• −(p + q)

Answer 1

In the given problem, we are given S_p = q and S_q = p

We need to find  S _p+q

Now, as we know,

`S_n = n/2 [2a + (n-1) d]`

So,

`S_p = p/2 [2a + (p - 1) d]`

  `q = p/2 [ 2a + ( p - 1)d]`

`2q = 2ap + p (p-1)d`                ............(1) 

Similarly,

`S_q = q/2 [ 2a + (q-1) d]`

  `p = q/2 [2a + (q-1)d]`

 `2p = 2ap + q(q-1)d`                ...............(2)

Subtracting (2) from (1), we get

2q - 2p = 2ap + [p ( p - 1) d ] - 2 aq - [q (q-1)d]

2q - 2 p = 2a (p-q) + [ p (p-1) - q(q-1)]d

-2(p-q) = 2a(p - q) + [(p² - q²) - ( p - q)] 

        -2 = 2a + ( p + q - 1 ) d                    ................(3) 

Now,

`S_(p+q) = (p+q)/2  [2a + (p+q - 1)d]`

`S_(p+q) = ((p+q))/2 (-2)`                 ........(Using 3) 

`S_(p+q) = - (p+q)`

Thus, `S_(p+q) = - (p+q)`