Question

20
Sum of first mterms of an AP is the same as the sum of its first in terms Show
that the sum of its first (min) terms is zero​

Answer 1

Step-by-step explanation:

Let a be the first term & d be the C.D of the AP

given

sum of 1st m term=sum of 1st n term

Sm=Sn

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

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

2am+m²-md=2an+n²-nd

2a(m-n)+d{m(m-1)+n(n-1l}=0

2a(m-n)+d{(m²-n²)-(m-n)}=0

2a(m-n)+d{(m-n)(m+n)-(m-n)}=0

2a(m-n)+d(m-n)(m+n-1)=0

(m-n){2a+d(m+n-1)}=0

2a+d(m+n-1)=0

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

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

=(m+n)/2[2a+(m+n-1)×{(-2a)/(m+n-1)}

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

=(m+n)/2×0

=0