Question

If Sm=n and Sn=m then Sm+n = ....,select a proper option (a), (b), (c) or (d) from given options so that the statement becomes correct.(All the problems refer to A.P.)
(a) -(m + n)
(b) 0
(c) m+n
(d) 2m–2n

Answer 1

Given, Sm = n and Sn = m
Let a is the first term and d is the common difference of an AP.
we know, $S_x=\frac{x}{2}[2a+(x-1)d]$
where x is the number of terms in AP.

now, Sm = n = m/2[2a + (m - 1)] ......(i)
Sn = m = n/2[2a + (n - 1)]d .....(ii)
and $S_{m+n}$=(m+n)/2[2a+(m+n-1)d].....(iii)

from eqs. (i) and (ii),
m - n = n/2[2a + (n - 1)d] - m/2[2a + (m -1)d]
(m - n) = an + n²d/2 - nd/2 - am - m²d/2 + md/2
(m -n) = -a(m - n) + d(m - n)/2 - (m² - n²)d/2
(m - n) = (m - n) [ -a + d/2 - (m + n)d/2]
1 = [-2a + d - (m + n)d]
2 = -[2a + (m + n - 1)d]
-2 = [2a + (m + n - 1)d]
multiplying (m + n)/2 both sides,
-2(m + n)/2 = (m + n)/2 [2a + (m + n - 1)d]
-(m + n) = $S_{m+n}$ [ from eq. (iii)

hence, option (a) is correct.

Answer 2

Answer:

option a is correct answer