Question

1. If the sum of m terms of an AP is n and the sum of n terms is m, then the sum of (m + n) terms is
(a) mn. (b) m + n
(c)2(m + n). (d) -(m + n)​

Answer 1

Answer:

• d) - ( m + n)

Solution :

Given sum of m terms = n

$\tt S_m = \dfrac{m}{2} [ 2 a + ( m -1) d]$

2 n = m [ 2 a + ( m -1) d]

2n = 2am + m ( m-1)d ____(1)

Sum of n terms = m

$\tt S_n = \dfrac{n}{2} [ 2a + ( n -1) d]$

2m = n [ 2a + ( n -1)d]

2m = 2an + n ( n -1)d _____(2)

Substracting (2) from (1)

2n - 2m = 2am + m ( m - 1) d - 2an - n( n-1)d

2 ( n - m ) = 2am+ m²d - md - 2an - n²d + nd

2 ( n - m) = 2am - 2 an + m²d - n²d - md + nd

2 ( n - m ) = 2a ( m - n) [ ( m² - n² ) - ( m - n )]d

2 ( n - m) = ( m - n) [ 2a + ( m + n - 1) d ]

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

2 ( n - m ) / - 1 ( n - m) = 2a + ( m + n -1) d

-2 = 2a + ( m + n -1) d _____(3)

Sum of m + n terms

$\tt S_{m + n} = \dfrac{m + n}{2} [ 2a + ( m + n -1) d]$

= m + n /2 × -2

= - ( m + n)